{"id":54,"date":"2021-12-15T09:53:07","date_gmt":"2021-12-15T09:53:07","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/erste-integrationsgesetze\/"},"modified":"2021-12-15T09:53:07","modified_gmt":"2021-12-15T09:53:07","slug":"erste-integrationsgesetze","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/erste-integrationsgesetze\/","title":{"raw":"Erste Integrationsgesetze","rendered":"Erste Integrationsgesetze"},"content":{"raw":"\n<style>.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: 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little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"zb4dab51b8527\" class=\"sectionHead\"><span class=\"titlemark\">4.3 <\/span> <a id=\"x1-1110003\"><\/a>Erste Integrationsgesetze<\/h3> <p class=\"noindent\">Wie schon zuvor betrachten wir hier Funktionen und den Begriff des Riemann-Integrals auf einem kompakten Intervall <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wir m\u00f6chten nun Eigenschaften des Riemann-Integrals nachweisen, die zu den Eigenschaften des Integrals von Treppenfunktionen (genauer Lemma <a href=\"..\/..\/chapter\/treppenfunktionen-und-deren-integral#x1-109006r7\">4.7<\/a> und Lemma <a href=\"..\/..\/chapter\/treppenfunktionen-und-deren-integral#x1-109009r8\">4.8<\/a>) analog sind. <a id=\"x1-111001r109\"><\/a> <\/p> <h4 id=\"z5215d76c5293\" class=\"subsectionHead\"><span class=\"titlemark\">4.3.1 <\/span> <a id=\"x1-1120001\"><\/a>Linearit\u00e4t<\/h4> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zfebb1c5d0cb8\"> <a id=\"x1-112001r19\"><\/a> <span class=\"ecbx-1095\">Satz 4.19 <\/span>(Linearit\u00e4t des Riemann-Integrals)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Die Menge<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>f<\/mi><mstyle class=\"text\"><mtext>&nbsp;ist&nbsp;Riemann-integrierbar<\/mtext><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">der Riemann-integrierbaren Funktionen auf <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">bildet einen Unterraum von <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und das<\/span> <span class=\"ecti-1095\">Integral ist eine lineare Funktion auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Das heisst, f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist<\/span> <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mi>s<\/mi><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <\/div> <p class=\"indent\">Im Beweis werden wir folgendes allgemeines Prinzip mehrmals anwenden. Falls <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>B<\/mi><\/math> nicht-leere Teilmengen von <math display=\"inline\"><mi>\u211d<\/mi><\/math> sind und <math display=\"inline\"><mi>B<\/mi><\/math> von oben beschr\u00e4nkt ist, dann ist <math display=\"inline\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> eine obere Schranke von <math display=\"inline\"><mi>A<\/mi><\/math> und daher <math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> (nach Definition des Supremums). Analog gilt <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> falls <math display=\"inline\"><mi>B<\/mi><\/math> von unten beschr\u00e4nkt ist. <\/p><p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Aus \u00dcbung <a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110010r14\">4.14<\/a> folgt die Inklusion <span class=\"maperiod\"><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi> <mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo><mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> sodass <math display=\"inline\"><mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> insbesondere nicht-leer ist. <\/p><p class=\"indent\">Sei nun <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> F\u00fcr Treppenfunktionen <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>o<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> mit <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>o<\/mi><\/math> gilt somit <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><mi>o<\/mi><\/math><\/span><span class=\"period\">.<\/span> Mit <math display=\"inline\"><mi>s<\/mi><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>u<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>s<\/mi><mi>u<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> und <math display=\"inline\"><mi>s<\/mi><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>o<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>s<\/mi><mi>o<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> nach Lemma <a href=\"..\/..\/chapter\/treppenfunktionen-und-deren-integral#x1-109006r7\">4.7<\/a> folgt <math display=\"inline\"><mi>s<\/mi><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> In der Tat ist                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>s<\/mi><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>s<\/mi><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>u<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><mi mathvariant=\"bold-script\">\u2131<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>s<\/mi><mi>u<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><mi mathvariant=\"bold-script\">\u2131<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>s<\/mi><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">eine Teilmenge von <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und analog f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Aus der Bemerkung vor dem Beweis folgt also <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Nach Proposition <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64007r62\">2.62<\/a> ist jedoch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>s<\/mi><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi><mi class=\"qopname\">inf<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Da aber <math display=\"inline\"><mi>f<\/mi><\/math> Riemann-integrierbar ist und somit <math display=\"inline\"><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> erf\u00fcllt ist, gilt in obiger Absch\u00e4tzung (wegen Gleichheit der kleinsten und der gr\u00f6ssten Zahl) \u00fcberall Gleichheit und wir schliessen <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Damit ist <math display=\"inline\"><mi>s<\/mi><mi>f<\/mi><\/math> Riemann-integrierbar mit Integral <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> Ist <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> so kehren sich in obigem alle Absch\u00e4tzungen, die <math display=\"inline\"><mi>s<\/mi><\/math> beinhalten, um (zum Beispiel gilt <math display=\"inline\"><mi>s<\/mi><mi>o<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><mi>u<\/mi><\/math>) und man erh\u00e4lt vollkommen analog die gew\u00fcnschte Aussage (siehe \u00dcbung <a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-112003r20\">4.20<\/a>). <\/p><p class=\"indent\">Wir zeigen nun Additivit\u00e4t des Integrals. Seien also <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> zwei Riemann-integrierbare Funktionen auf <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> und <math display=\"inline\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>o<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>o<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> Treppenfunktionen mit                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>o<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>o<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dann ist auch <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>o<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>o<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> was gem\u00e4ss Lemma <a href=\"..\/..\/chapter\/treppenfunktionen-und-deren-integral#x1-109006r7\">4.7<\/a> <\/p><table id=\"z63b1cd258a8f\" class=\"equation\"><tr><td> <a id=\"x1-112002r3\"><\/a> <math class=\"equation\" display=\"block\"> <mtable class=\"aligned\"><mtr><mtd columnalign=\"right\"><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd columnalign=\"left\"> <mo class=\"MathClass-rel\">\u2286<\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><\/mtd><mtd columnalign=\"right\" \/> <\/mtr><mtr><mtd columnalign=\"right\"><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mtd><mtd columnalign=\"left\"> <mo class=\"MathClass-rel\">\u2286<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mtd> <\/mtr> <\/mtable> <\/math><\/td><td class=\"eq-no\">(4.3)<\/td><\/tr><\/table> <p class=\"indent\">zur Folge hat. Des Weiteren gilt nach Proposition <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64008r63\">2.63<\/a>, dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">nach Riemann-Integrierbarkeit von <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> und <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> und ebenso                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Gemeinsam mit der Bemerkung vor dem Beweis ergibt sich nun wiederum <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dies zeigt                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und insbesondere Riemann-Integrierbarkeit von <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Wir haben also die Linearit\u00e4t des Riemann-Integrals bewiesen. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z16c3f5d10942\"> <a id=\"x1-112003r20\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.20 <\/span>(Negative Vielfache)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Formulieren Sie den Fall<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">im obigen Beweis aus.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zb87c942daa9d\"> <a id=\"x1-112004r21\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.21.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass Gleichheit in<\/span> (<a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-112002r3\">4.3<\/a>) <span class=\"ecti-1095\">(siehe obigen Beweis) nicht erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt sein muss.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Verwenden                      Sie                      die                      Polynome<\/span> <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110011r15\"><span class=\"ecti-1095\">4.15<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z04e53b283fcf\"> <a id=\"x1-112005r22\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.22 <\/span>(\u00c4ndern bei einem Punkt)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">Riemann-integrierbar.                                                                                   Sei<\/span> <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2217<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">eine Funktion,       die       erhalten       wurde,       indem       der       Wert       von<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">an                      nur                      einem                      Punkt                      in<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">abge<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndert                  wurde.                   Zeigen                  Sie,                   dass<\/span> <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2217<\/mo> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">Riemann-integrierbar       ist       und       das       gleiche       Riemann-Integral       wie<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">hat.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Verwenden Sie Satz <\/span><a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-112001r19\"><span class=\"ecti-1095\">4.19<\/span><\/a> <span class=\"ecti-1095\">und f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">die<\/span> <span class=\"ecti-1095\">Treppenfunktion <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gegeben durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>t<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><mi>c<\/mi><\/mtd><mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>0<\/mn><\/mtd><mtd class=\"array\" columnalign=\"center\"> <mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <a id=\"x1-112006r112\"><\/a> <h4 id=\"ze757ce7484d8\" class=\"subsectionHead\"><span class=\"titlemark\">4.3.2 <\/span> <a id=\"x1-1130002\"><\/a>Monotonie<\/h4> <p class=\"noindent\">F\u00fcr <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> definieren wir Funktionen <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> durch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> Die Funktion <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup><\/math> ist der <span class=\"ecbx-1095\">Positivteil <\/span>von <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msup> <\/math> ist der <span class=\"ecbx-1095\">Negativteil <\/span>von <math display=\"inline\"><mi>f<\/mi><\/math> und <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> ist der <span class=\"ecbx-1095\">Absolutbetrag <\/span>von <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z5fa57ce5a79a\"> <a id=\"x1-113001r23\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.23 <\/span>(Eigenschaften vom Positiv- und Negativteil)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie die Gleichungen<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"ze0bec821e9e3\"> <a id=\"x1-113002r24\"><\/a> <span class=\"ecbx-1095\">Satz 4.24 <\/span>(Monotonie des Riemann-Integrals)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r zwei Funktionen <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gelten folgende Monotonie-Eigenschaften des Riemann-Integrals:<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ist, so gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist, so gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Funktion <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/math><span class=\"ecti-1095\">ist<\/span> <span class=\"ecti-1095\">Riemann-integrierbar auf <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">und es gilt die <\/span><span class=\"ecbi-1095\">Dreiecksungleichung<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">|<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/dd><\/dl> <\/div> <p class=\"indent\">Wir m\u00f6chten kurz erkl\u00e4ren, wieso sich die Ungleichung in Punkt (iii) des obigen Satzes Dreiecksungleichung nennt. Tats\u00e4chlich sieht man kein Dreieck, im Gegensatz zur Dreiecksungleichung                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">,<\/span> die geometrisch direkt begr\u00fcndet werden kann (wie?). Es gilt auch die verallgemeinerte Dreiecksungleichung <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>i<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>z<\/mi><\/mrow><mrow> <mi>i<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>i<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow> <mi>i<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">,<\/span> wie man direkt aus der Dreiecksungleichung und vollst\u00e4ndiger Induktion folgern kann (siehe \u00dcbung <a href=\"..\/..\/chapter\/summen-und-produkte#x1-78002r4\">3.4<\/a>). Die Aussage (iii) in Satz <a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-113002r24\">4.24<\/a> ist eine \u201ekontinuierliche Version\u201c der verallgemeinerten Dreiecksungleichung, weswegen wir von der Dreiecksungleichung f\u00fcr das Riemann-Integral sprechen. <\/p> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=760&amp;url=Pictures\/R-integral\/triineq\/triineq.pdf\"><img id=\"z27a86e9c9dc2\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/R-integral\/triineq\/triineq.svg\" width=\"760\"><\/div> <a id=\"x1-113006r3\"><\/a> <a id=\"x1-113007\"><\/a> <br><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;4.3:    <\/span><span class=\"content\">Wir    sehen    hier    den    Graphen    einer    Funktion               <math display=\"inline\"><mi>f<\/mi><\/math>             links            und            der            entsprechenden            Funkton               <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math>       rechts.                                    Dabei                                   stellt               <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math>             einen                            Nettofl\u00e4cheninhalt                            und               <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">|<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math>             einen Fl\u00e4cheninhalt dar.                                                            &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">F\u00fcr <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> wie in (i) ist die konstante Funktion <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> eine Treppenfunktion mit <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> und <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>0<\/mn> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>u<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/munderover><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">folgt. <\/p><p class=\"indent\">Falls <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> wie in (ii) gilt, so ist <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> und                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">nach Linearit\u00e4t des Riemann-Integrals (Satz <a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-112001r19\">4.19<\/a>) und Teil (i). Dies zeigt (ii). <\/p><p class=\"indent\">F\u00fcr (iii) wollen wir zuerst zeigen, dass f\u00fcr ein <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> auch <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <\/math> Riemann-integrierbar ist. Dazu bemerken wir zuerst, dass f\u00fcr <math display=\"inline\"><mi>s<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> die Ungleichung <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>t<\/mi><\/math> impliziert, dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>s<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>s<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <msup><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mstyle class=\"mbox\"><mtext>&nbsp;und&nbsp;<\/mtext><\/mstyle><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>s<\/mi><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dies ergibt sich aus der Unterscheidung der F\u00e4lle <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>t<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>t<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>s<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>t<\/mi><\/math><\/span><span class=\"period\">.<\/span> <button class=\"hover-trigger\">(Wieso?)<\/button><span class=\"hover-text\"><span class=\"marginpar\">Falls <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>t<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>0<\/mn><\/math> dann ist <math display=\"inline\"><msup><mrow><mi>s<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> und <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>s<\/mi><\/math><\/span><span class=\"period\">.<\/span> Falls <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>t<\/mi><\/math> dann ist <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>s<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>t<\/mi><\/math><\/span><span class=\"period\">,<\/span> und damit <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>t<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>s<\/mi><\/math><\/span><span class=\"period\">.<\/span> Falls <span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>s<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>t<\/mi><\/math><\/span><span class=\"period\">,<\/span> dann ist <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>s<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msup><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>t<\/mi><\/math> und                                                                                                                                                                           <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>s<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/span><\/span> Da <math display=\"inline\"><mi>f<\/mi><\/math> Riemann-integrierbar ist, gibt es nach Proposition <a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110005r12\">4.12<\/a> (iii) zu jedem <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> zwei Treppenfunktion <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>o<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> mit <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>o<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><mi>o<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span> Verkn\u00fcpfen wir diese mit der Funktion <span class=\"maperiod\"><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> so ergibt sich <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <msup><mrow><mi>o<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msup><mrow><mi>o<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>u<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>o<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>u<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und daher nach (ii) auch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msup><mrow><mi>o<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>u<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>o<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Allerdings sind <math display=\"inline\"><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><msup><mrow><mi>o<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup><\/math> wieder Treppenfunktionen. Nach der dritten Charakterisierung in Proposition <a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110005r12\">4.12<\/a> ergibt sich somit, dass <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <\/math> Riemann-integrierbar ist, da <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig war.                                                                                                                                                                           <\/p><p class=\"indent\">Mittels Satz <a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-112001r19\">4.19<\/a> erhalten wir, dass <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><\/math> auch Riemann-integrierbar ist. Aus <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> und <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> folgt aus (ii) nun <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">|<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">|<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was zur Dreiecksungleichung \u00e4quivalent ist. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z7111ff18cd0b\"> <a id=\"x1-113008r25\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.25 <\/span>(Modifizierte Dirichlet- oder Riemann-Funktion)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass die Funktion<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2192<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mn>0<\/mn><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi><mstyle class=\"text\"><mtext>&nbsp;irrational<\/mtext><\/mstyle> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>q<\/mi><\/mrow><\/mfrac><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>p<\/mi><\/mrow> <mrow><mi>q<\/mi><\/mrow><\/mfrac><mstyle class=\"text\"><mtext>&nbsp;mit&nbsp;<\/mtext><\/mstyle><mi>p<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>q<\/mi><mstyle class=\"text\"><mtext>&nbsp;teilerfremd<\/mtext><\/mstyle><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Riemann-integrierbar ist. Als Hilfestellung stellen wir den Graphen dar, aber <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berlassen Ihnen die<\/span> <span class=\"ecti-1095\">Interpretation des Graphen und die sich daraus ergebenden <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">berlegungen. <\/span><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"339px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/BT59E6PF\/width\/688\/height\/339\/border\/888888\/rc\/false\/ai\/false\/sdz\/true\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"> <\/p> <\/div> <a id=\"x1-113009r113\"><\/a> <h4 id=\"zbbea64040171\" class=\"subsectionHead\"><span class=\"titlemark\">4.3.3 <\/span> <a id=\"x1-1140003\"><\/a>Teilintervalle<\/h4> <p class=\"noindent\">Es seien <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>c<\/mi><\/math> drei reelle Zahlen. Dann definiert eine Funktion <math display=\"inline\"><mi>f<\/mi><\/math> auf dem Intervall <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> die Funktion <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/msub><\/math> auf <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> und die Funktion <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/msub><\/math> auf <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> Dabei gilt <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Umgekehrt k\u00f6nnen wir Funktionen <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> mit <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> verwenden, um eine Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> zu definieren. In diesem Sinne entspricht die Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> zwei Funktionen <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zbea0736bc156\"> <a id=\"x1-114001r26\"><\/a> <span class=\"ecbx-1095\">Satz 4.26 <\/span>(Additionseigenschaft bez\u00fcglich Intervallen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Unter Verwendung obiger Notation gilt, dass<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">genau dann<\/span> <span class=\"ecti-1095\">Riemann-integrierbar ist, wenn <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">Riemann-integrierbar sind. In diesem Fall ist<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir verifizieren zuerst die behauptete Formel f\u00fcr Treppenfunktionen. Dazu betrachten wir eine Treppenfunktion <math display=\"inline\"><mi>t<\/mi><\/math> auf <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> und eine Zerlegung in Konstanzintervalle von <math display=\"inline\"><mi>t<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u2128<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>c<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dabei d\u00fcrfen wir wegen Lemma <a href=\"..\/..\/chapter\/treppenfunktionen-und-deren-integral#x1-109004r5\">4.5<\/a> ohne Beschr\u00e4nkung der Allgemeinheit annehmen, dass <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>m<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi><\/math> f\u00fcr ein <span class=\"maperiod\"><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> F\u00fcr <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> sei <math display=\"inline\"><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/math> der Konstanzwert von <math display=\"inline\"><mi>t<\/mi><\/math> auf <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span>                                                                                                                                                                           Dann gilt <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>t<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>c<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mi>\u0394<\/mi><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-114002r4\" \/><mstyle class=\"maketag\"><mtext>(4.4)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>m<\/mi><\/mrow><\/munderover><msub><mrow><mi>c<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mi>\u0394<\/mi><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>c<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mi>\u0394<\/mi><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>t<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>t<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow> <mo class=\"MathClass-open\">[<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-114003r5\" \/><mstyle class=\"maketag\"><mtext>(4.5)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Sei <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> eine Funktion und definiere <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Gegeben <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi> <mi mathvariant=\"bold-script\">\u2131<\/mi> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> mit <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><\/math> kann man ebenso <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>u<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>u<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo> <\/mrow> <\/msub> <\/math> definieren. Es gilt <math display=\"inline\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> und <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Wegen Gleichung (<a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114002r4\">4.4<\/a>) erhalten wir, dass <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>u<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>u<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><msub><mrow><mi>u<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-114004r6\" \/><mstyle class=\"maketag\"><mtext>(4.6)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was wiederum <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> zur Folge hat. Umgekehrt kann man, gegeben Treppenfunktionen <math display=\"inline\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> mit <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> eine Treppenfunktion <math display=\"inline\"><mi>u<\/mi><\/math> auf <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> definieren, die ebenso <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><\/math> und Gleichung (<a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114004r6\">4.6<\/a>) erf\u00fcllt. <button class=\"hover-trigger\">(Wie genau?)<\/button><span class=\"hover-text\"><span class=\"marginpar\">Wegen Lemma <a href=\"..\/..\/chapter\/treppenfunktionen-und-deren-integral#x1-109004r5\">4.5<\/a> spielt der Funktionswert einer Treppenfunktion bei einem einzelnen Wert keine Rolle. Deswegen k\u00f6nnen wir beispielsweise die Funktion <math display=\"inline\"><mi>u<\/mi><\/math> definiert durch <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> verwenden.<\/span><\/span> Dadurch ist <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und wegen der Additionseigenschaft des Supremums in Proposition <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64008r63\">2.63<\/a> gilt <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-114005r7\" \/><mstyle class=\"maketag\"><mtext>(4.7)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Analog zeigt man, dass <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-114006r8\" \/><mstyle class=\"maketag\"><mtext>(4.8)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Ist nun <math display=\"inline\"><mi>f<\/mi><\/math> Riemann-integrierbar, dann ist <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2265<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2265<\/mo><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">\u00dcberall in dieser Kette von Ungleichungen gilt also Gleichheit. Somit ist <math display=\"inline\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">=<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi> <\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und dadurch auch <span class=\"maperiod\"><math display=\"inline\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi> <\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Das heisst, dass <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> und <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> Riemann-integrierbar sind und Gleichung&nbsp;(<a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114005r7\">4.7<\/a>) wird zur gew\u00fcnschten Additionseigenschaft f\u00fcr das Riemann-Integral. <\/p><p class=\"indent\">Falls <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> Riemann-integrierbar sind, dann gilt <math display=\"inline\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi> <\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Dies impliziert gemeinsam mit den Gleichungen&nbsp;(<a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114005r7\">4.7<\/a>), (<a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114006r8\">4.8<\/a>) auch <math display=\"inline\"><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und die Additionseigenschaft. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Sei <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> ein kompaktes Intervall mit <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> Ist <math display=\"inline\"><mi>f<\/mi><\/math> eine Funktion, die auf einer gr\u00f6sseren Menge als <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> definiert ist, so werden wir anstelle von <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/msub><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> trotzdem meist <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> schreiben, wenn <math display=\"inline\"><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/msub><\/math> Riemann-integrierbar ist. Auch definieren wir die folgende Erweiterung des Riemann-Integrals <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>a<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace class=\"quad\" width=\"1em\" \/><mstyle class=\"text\"><mtext>und<\/mtext><\/mstyle><mspace class=\"quad\" width=\"1em\" \/><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>a<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-114007r9\" \/><mstyle class=\"maketag\"><mtext>(4.9)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Diese Definition vereinfacht die Notation und macht auf Grund der Aussage in folgender \u00dcbung Sinn. <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z91051fb2f94b\"> <a id=\"x1-114008r27\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.27 <\/span>(Intervalladditivit\u00e4t)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>I<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ein kompaktes Intervall und sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>I<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie die Additionseigenschaft in Satz <\/span><a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114001r26\"><span class=\"ecti-1095\">4.26<\/span><\/a> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>I<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Wir unterscheiden F<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lle abh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngig von der Anordnung der Punkte<\/span> <math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>c<\/mi><\/math> <span class=\"ecti-1095\">im<\/span> <span class=\"ecti-1095\">Intervall <\/span><math display=\"inline\"><mi>I<\/mi><\/math> <span class=\"ecti-1095\">und zeigen jeweils, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-114009r10\" \/><mstyle class=\"maketag\"><mtext>(4.10)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">gilt wie gew<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nscht.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">1.<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Angenommen <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>c<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann ist<\/span> (<a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114009r10\">4.10<\/a>) <span class=\"ecti-1095\">genau die Additionseigenschaft aus Satz <\/span><a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114001r26\"><span class=\"ecti-1095\">4.26<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">2.<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Angenommen <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann ist <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">per Definition und es gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">3.<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>c<\/mi><\/math> <span class=\"ecti-1095\">gilt, so geht man wie in vorherigem Fall vor.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">4.<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Angenommen es gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>c<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann ist nach Satz <\/span><a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114001r26\"><span class=\"ecti-1095\">4.26<\/span><\/a> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>a<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Somit gilt nach den Definitionen vor dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>a<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">5.<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die F<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>c<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>c<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>a<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>a<\/mi><\/math> <span class=\"ecti-1095\">werden <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hnlich behandelt.<\/span><\/dd><\/dl> <p class=\"noindent\"><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z867f5c36b447\"> <a id=\"x1-114015r28\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.28 <\/span>(Stetigkeit des partikul\u00e4ren Integrals)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">und<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Riemann-integrierbare Funktion. Zeigen Sie, dass das sogenannte partikul<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re Integral<\/span> <\/p><table id=\"zb31a59c3a74f\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <\/math><\/td><\/tr><\/table> <p class=\"indent\"><span class=\"ecti-1095\">eine stetige reellwertige Funktion auf<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">definiert. Ist diese Funktion auch gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssig oder Lipschitz-stetig (siehe <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/stetige-funktionen-auf-kompakten-intervallen#x1-102015r80\"><span class=\"ecti-1095\">3.80<\/span><\/a> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">letzteren Begriff)? <\/span><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"565px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/vvfnpyzd\/width\/688\/height\/565\/border\/888888\/rc\/false\/ai\/false\/sdz\/true\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"> <\/p> <\/div> <a id=\"x1-114016r111\"><\/a> \n","rendered":"\n<style 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-65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"zb4dab51b8527\" class=\"sectionHead\"><span class=\"titlemark\">4.3 <\/span> <a id=\"x1-1110003\"><\/a>Erste Integrationsgesetze<\/h3> <p class=\"noindent\">Wie schon zuvor betrachten wir hier Funktionen und den Begriff des Riemann-Integrals auf einem kompakten Intervall <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wir m\u00f6chten nun Eigenschaften des Riemann-Integrals nachweisen, die zu den Eigenschaften des Integrals von Treppenfunktionen (genauer Lemma <a href=\"..\/..\/chapter\/treppenfunktionen-und-deren-integral#x1-109006r7\">4.7<\/a> und Lemma <a href=\"..\/..\/chapter\/treppenfunktionen-und-deren-integral#x1-109009r8\">4.8<\/a>) analog sind. <a id=\"x1-111001r109\"><\/a> <\/p> <h4 id=\"z5215d76c5293\" class=\"subsectionHead\"><span class=\"titlemark\">4.3.1 <\/span> <a id=\"x1-1120001\"><\/a>Linearit\u00e4t<\/h4> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zfebb1c5d0cb8\"> <a id=\"x1-112001r19\"><\/a> <span class=\"ecbx-1095\">Satz 4.19 <\/span>(Linearit\u00e4t des Riemann-Integrals)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Die Menge<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>f<\/mi><mstyle class=\"text\"><mtext>&nbsp;ist&nbsp;Riemann-integrierbar<\/mtext><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">der Riemann-integrierbaren Funktionen auf <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">bildet einen Unterraum von <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und das<\/span> <span class=\"ecti-1095\">Integral ist eine lineare Funktion auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Das heisst, f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist<\/span> <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mi>s<\/mi><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <\/div> <p class=\"indent\">Im Beweis werden wir folgendes allgemeines Prinzip mehrmals anwenden. Falls <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>B<\/mi><\/math> nicht-leere Teilmengen von <math display=\"inline\"><mi>\u211d<\/mi><\/math> sind und <math display=\"inline\"><mi>B<\/mi><\/math> von oben beschr\u00e4nkt ist, dann ist <math display=\"inline\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> eine obere Schranke von <math display=\"inline\"><mi>A<\/mi><\/math> und daher <math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> (nach Definition des Supremums). Analog gilt <span class=\"maperiod\"><math display=\"inline\"><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>B<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> falls <math display=\"inline\"><mi>B<\/mi><\/math> von unten beschr\u00e4nkt ist. <\/p><div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Aus \u00dcbung <a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110010r14\">4.14<\/a> folgt die Inklusion <span class=\"maperiod\"><math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi> <mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo><mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> sodass <math display=\"inline\"><mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> insbesondere nicht-leer ist. <\/p><p class=\"indent\">Sei nun <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> F\u00fcr Treppenfunktionen <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>o<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> mit <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>o<\/mi><\/math> gilt somit <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><mi>o<\/mi><\/math><\/span><span class=\"period\">.<\/span> Mit <math display=\"inline\"><mi>s<\/mi><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>u<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>s<\/mi><mi>u<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> und <math display=\"inline\"><mi>s<\/mi><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>o<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>s<\/mi><mi>o<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> nach Lemma <a href=\"..\/..\/chapter\/treppenfunktionen-und-deren-integral#x1-109006r7\">4.7<\/a> folgt <math display=\"inline\"><mi>s<\/mi><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> In der Tat ist                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>s<\/mi><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>s<\/mi><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>u<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><mi mathvariant=\"bold-script\">\u2131<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>s<\/mi><mi>u<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><mi mathvariant=\"bold-script\">\u2131<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>s<\/mi><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">eine Teilmenge von <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und analog f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Aus der Bemerkung vor dem Beweis folgt also <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Nach Proposition <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64007r62\">2.62<\/a> ist jedoch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>s<\/mi><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi><mi class=\"qopname\">inf<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Da aber <math display=\"inline\"><mi>f<\/mi><\/math> Riemann-integrierbar ist und somit <math display=\"inline\"><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> erf\u00fcllt ist, gilt in obiger Absch\u00e4tzung (wegen Gleichheit der kleinsten und der gr\u00f6ssten Zahl) \u00fcberall Gleichheit und wir schliessen <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Damit ist <math display=\"inline\"><mi>s<\/mi><mi>f<\/mi><\/math> Riemann-integrierbar mit Integral <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> Ist <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> so kehren sich in obigem alle Absch\u00e4tzungen, die <math display=\"inline\"><mi>s<\/mi><\/math> beinhalten, um (zum Beispiel gilt <math display=\"inline\"><mi>s<\/mi><mi>o<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>s<\/mi><mi>u<\/mi><\/math>) und man erh\u00e4lt vollkommen analog die gew\u00fcnschte Aussage (siehe \u00dcbung <a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-112003r20\">4.20<\/a>). <\/p><p class=\"indent\">Wir zeigen nun Additivit\u00e4t des Integrals. Seien also <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> zwei Riemann-integrierbare Funktionen auf <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> und <math display=\"inline\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>o<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>o<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> Treppenfunktionen mit                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>o<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>o<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dann ist auch <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>o<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>o<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> was gem\u00e4ss Lemma <a href=\"..\/..\/chapter\/treppenfunktionen-und-deren-integral#x1-109006r7\">4.7<\/a> <\/p><table id=\"z63b1cd258a8f\" class=\"equation\"><tr><td> <a id=\"x1-112002r3\"><\/a> <math class=\"equation\" display=\"block\"> <mtable class=\"aligned\"><mtr><mtd columnalign=\"right\"><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd columnalign=\"left\"> <mo class=\"MathClass-rel\">\u2286<\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><\/mtd><mtd columnalign=\"right\" \/> <\/mtr><mtr><mtd columnalign=\"right\"><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mtd><mtd columnalign=\"left\"> <mo class=\"MathClass-rel\">\u2286<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mtd> <\/mtr> <\/mtable> <\/math><\/td><td class=\"eq-no\">(4.3)<\/td><\/tr><\/table> <p class=\"indent\">zur Folge hat. Des Weiteren gilt nach Proposition <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64008r63\">2.63<\/a>, dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">nach Riemann-Integrierbarkeit von <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> und <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> und ebenso                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Gemeinsam mit der Bemerkung vor dem Beweis ergibt sich nun wiederum <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dies zeigt                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und insbesondere Riemann-Integrierbarkeit von <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Wir haben also die Linearit\u00e4t des Riemann-Integrals bewiesen. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z16c3f5d10942\"> <a id=\"x1-112003r20\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.20 <\/span>(Negative Vielfache)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Formulieren Sie den Fall<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">im obigen Beweis aus.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zb87c942daa9d\"> <a id=\"x1-112004r21\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.21.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass Gleichheit in<\/span> (<a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-112002r3\">4.3<\/a>) <span class=\"ecti-1095\">(siehe obigen Beweis) nicht erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt sein muss.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Verwenden                      Sie                      die                      Polynome<\/span> <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110011r15\"><span class=\"ecti-1095\">4.15<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z04e53b283fcf\"> <a id=\"x1-112005r22\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.22 <\/span>(\u00c4ndern bei einem Punkt)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">Riemann-integrierbar.                                                                                   Sei<\/span> <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2217<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">eine Funktion,       die       erhalten       wurde,       indem       der       Wert       von<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">an                      nur                      einem                      Punkt                      in<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">abge<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndert                  wurde.                   Zeigen                  Sie,                   dass<\/span> <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2217<\/mo> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">Riemann-integrierbar       ist       und       das       gleiche       Riemann-Integral       wie<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">hat.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Verwenden Sie Satz <\/span><a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-112001r19\"><span class=\"ecti-1095\">4.19<\/span><\/a> <span class=\"ecti-1095\">und f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">die<\/span> <span class=\"ecti-1095\">Treppenfunktion <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gegeben durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>t<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><mi>c<\/mi><\/mtd><mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>0<\/mn><\/mtd><mtd class=\"array\" columnalign=\"center\"> <mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <a id=\"x1-112006r112\"><\/a> <h4 id=\"ze757ce7484d8\" class=\"subsectionHead\"><span class=\"titlemark\">4.3.2 <\/span> <a id=\"x1-1130002\"><\/a>Monotonie<\/h4> <p class=\"noindent\">F\u00fcr <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> definieren wir Funktionen <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> durch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> Die Funktion <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup><\/math> ist der <span class=\"ecbx-1095\">Positivteil <\/span>von <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msup> <\/math> ist der <span class=\"ecbx-1095\">Negativteil <\/span>von <math display=\"inline\"><mi>f<\/mi><\/math> und <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> ist der <span class=\"ecbx-1095\">Absolutbetrag <\/span>von <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z5fa57ce5a79a\"> <a id=\"x1-113001r23\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.23 <\/span>(Eigenschaften vom Positiv- und Negativteil)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie die Gleichungen<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"ze0bec821e9e3\"> <a id=\"x1-113002r24\"><\/a> <span class=\"ecbx-1095\">Satz 4.24 <\/span>(Monotonie des Riemann-Integrals)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r zwei Funktionen <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gelten folgende Monotonie-Eigenschaften des Riemann-Integrals:<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ist, so gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist, so gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Funktion <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/math><span class=\"ecti-1095\">ist<\/span> <span class=\"ecti-1095\">Riemann-integrierbar auf <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">und es gilt die <\/span><span class=\"ecbi-1095\">Dreiecksungleichung<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">|<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/dd><\/dl> <\/div> <p class=\"indent\">Wir m\u00f6chten kurz erkl\u00e4ren, wieso sich die Ungleichung in Punkt (iii) des obigen Satzes Dreiecksungleichung nennt. Tats\u00e4chlich sieht man kein Dreieck, im Gegensatz zur Dreiecksungleichung                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">,<\/span> die geometrisch direkt begr\u00fcndet werden kann (wie?). Es gilt auch die verallgemeinerte Dreiecksungleichung <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>i<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>z<\/mi><\/mrow><mrow> <mi>i<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>i<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow> <mi>i<\/mi><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">,<\/span> wie man direkt aus der Dreiecksungleichung und vollst\u00e4ndiger Induktion folgern kann (siehe \u00dcbung <a href=\"..\/..\/chapter\/summen-und-produkte#x1-78002r4\">3.4<\/a>). Die Aussage (iii) in Satz <a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-113002r24\">4.24<\/a> ist eine \u201ekontinuierliche Version\u201c der verallgemeinerten Dreiecksungleichung, weswegen wir von der Dreiecksungleichung f\u00fcr das Riemann-Integral sprechen. <\/p> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=760&amp;url=Pictures\/R-integral\/triineq\/triineq.pdf\"><img decoding=\"async\" id=\"z27a86e9c9dc2\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/R-integral\/triineq\/triineq.svg\" width=\"760\" \/><\/div> <a id=\"x1-113006r3\"><\/a> <a id=\"x1-113007\"><\/a> <br \/><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;4.3:    <\/span><span class=\"content\">Wir    sehen    hier    den    Graphen    einer    Funktion               <math display=\"inline\"><mi>f<\/mi><\/math>             links            und            der            entsprechenden            Funkton               <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math>       rechts.                                    Dabei                                   stellt               <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math>             einen                            Nettofl\u00e4cheninhalt                            und               <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">|<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math>             einen Fl\u00e4cheninhalt dar.                                                            &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">F\u00fcr <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> wie in (i) ist die konstante Funktion <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> eine Treppenfunktion mit <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> und <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mn>0<\/mn> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>u<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/munderover><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">folgt. <\/p><p class=\"indent\">Falls <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> wie in (ii) gilt, so ist <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> und                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">nach Linearit\u00e4t des Riemann-Integrals (Satz <a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-112001r19\">4.19<\/a>) und Teil (i). Dies zeigt (ii). <\/p><p class=\"indent\">F\u00fcr (iii) wollen wir zuerst zeigen, dass f\u00fcr ein <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> auch <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <\/math> Riemann-integrierbar ist. Dazu bemerken wir zuerst, dass f\u00fcr <math display=\"inline\"><mi>s<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> die Ungleichung <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>t<\/mi><\/math> impliziert, dass <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>s<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>s<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <msup><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mstyle class=\"mbox\"><mtext>&nbsp;und&nbsp;<\/mtext><\/mstyle><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>s<\/mi><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dies ergibt sich aus der Unterscheidung der F\u00e4lle <span class=\"maperiod\"><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>t<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>t<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>s<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>t<\/mi><\/math><\/span><span class=\"period\">.<\/span> <button class=\"hover-trigger\">(Wieso?)<\/button><span class=\"hover-text\"><span class=\"marginpar\">Falls <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>t<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>0<\/mn><\/math> dann ist <math display=\"inline\"><msup><mrow><mi>s<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> und <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>s<\/mi><\/math><\/span><span class=\"period\">.<\/span> Falls <math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>t<\/mi><\/math> dann ist <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>s<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>t<\/mi><\/math><\/span><span class=\"period\">,<\/span> und damit <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>t<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>s<\/mi><\/math><\/span><span class=\"period\">.<\/span> Falls <span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>s<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>t<\/mi><\/math><\/span><span class=\"period\">,<\/span> dann ist <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>s<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msup><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>t<\/mi><\/math> und                                                                                                                                                                           <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>s<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>s<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/span><\/span> Da <math display=\"inline\"><mi>f<\/mi><\/math> Riemann-integrierbar ist, gibt es nach Proposition <a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110005r12\">4.12<\/a> (iii) zu jedem <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> zwei Treppenfunktion <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>o<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> mit <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>o<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><mi>o<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>u<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span> Verkn\u00fcpfen wir diese mit der Funktion <span class=\"maperiod\"><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> so ergibt sich <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <msup><mrow><mi>o<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msup><mrow><mi>o<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>u<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>o<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>u<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und daher nach (ii) auch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msup><mrow><mi>o<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>u<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>o<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>u<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Allerdings sind <math display=\"inline\"><msup><mrow><mi>u<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><msup><mrow><mi>o<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup><\/math> wieder Treppenfunktionen. Nach der dritten Charakterisierung in Proposition <a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110005r12\">4.12<\/a> ergibt sich somit, dass <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msup> <\/math> Riemann-integrierbar ist, da <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig war.                                                                                                                                                                           <\/p><p class=\"indent\">Mittels Satz <a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-112001r19\">4.19<\/a> erhalten wir, dass <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><\/math> auch Riemann-integrierbar ist. Aus <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> und <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> folgt aus (ii) nun <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">|<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">|<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was zur Dreiecksungleichung \u00e4quivalent ist. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z7111ff18cd0b\"> <a id=\"x1-113008r25\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.25 <\/span>(Modifizierte Dirichlet- oder Riemann-Funktion)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass die Funktion<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2192<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mn>0<\/mn><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi><mstyle class=\"text\"><mtext>&nbsp;irrational<\/mtext><\/mstyle> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>q<\/mi><\/mrow><\/mfrac><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>p<\/mi><\/mrow> <mrow><mi>q<\/mi><\/mrow><\/mfrac><mstyle class=\"text\"><mtext>&nbsp;mit&nbsp;<\/mtext><\/mstyle><mi>p<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>q<\/mi><mstyle class=\"text\"><mtext>&nbsp;teilerfremd<\/mtext><\/mstyle><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Riemann-integrierbar ist. Als Hilfestellung stellen wir den Graphen dar, aber <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berlassen Ihnen die<\/span> <span class=\"ecti-1095\">Interpretation des Graphen und die sich daraus ergebenden <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">berlegungen. <\/span><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"339px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/BT59E6PF\/width\/688\/height\/339\/border\/888888\/rc\/false\/ai\/false\/sdz\/true\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><div class=\"wp-nocaption \"><\/div> <\/div> <a id=\"x1-113009r113\"><\/a> <h4 id=\"zbbea64040171\" class=\"subsectionHead\"><span class=\"titlemark\">4.3.3 <\/span> <a id=\"x1-1140003\"><\/a>Teilintervalle<\/h4> <p class=\"noindent\">Es seien <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>c<\/mi><\/math> drei reelle Zahlen. Dann definiert eine Funktion <math display=\"inline\"><mi>f<\/mi><\/math> auf dem Intervall <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> die Funktion <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/msub><\/math> auf <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> und die Funktion <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/msub><\/math> auf <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> Dabei gilt <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Umgekehrt k\u00f6nnen wir Funktionen <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> mit <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> verwenden, um eine Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> zu definieren. In diesem Sinne entspricht die Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> zwei Funktionen <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zbea0736bc156\"> <a id=\"x1-114001r26\"><\/a> <span class=\"ecbx-1095\">Satz 4.26 <\/span>(Additionseigenschaft bez\u00fcglich Intervallen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Unter Verwendung obiger Notation gilt, dass<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">genau dann<\/span> <span class=\"ecti-1095\">Riemann-integrierbar ist, wenn <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">Riemann-integrierbar sind. In diesem Fall ist<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir verifizieren zuerst die behauptete Formel f\u00fcr Treppenfunktionen. Dazu betrachten wir eine Treppenfunktion <math display=\"inline\"><mi>t<\/mi><\/math> auf <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> und eine Zerlegung in Konstanzintervalle von <math display=\"inline\"><mi>t<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u2128<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>c<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dabei d\u00fcrfen wir wegen Lemma <a href=\"..\/..\/chapter\/treppenfunktionen-und-deren-integral#x1-109004r5\">4.5<\/a> ohne Beschr\u00e4nkung der Allgemeinheit annehmen, dass <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>m<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi><\/math> f\u00fcr ein <span class=\"maperiod\"><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> F\u00fcr <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> sei <math display=\"inline\"><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/math> der Konstanzwert von <math display=\"inline\"><mi>t<\/mi><\/math> auf <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span>                                                                                                                                                                           Dann gilt <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>t<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>c<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mi>\u0394<\/mi><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-114002r4\" \/><mstyle class=\"maketag\"><mtext>(4.4)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>m<\/mi><\/mrow><\/munderover><msub><mrow><mi>c<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mi>\u0394<\/mi><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mi>m<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msub><mrow><mi>c<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mi>\u0394<\/mi><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>t<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>t<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow> <mo class=\"MathClass-open\">[<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-114003r5\" \/><mstyle class=\"maketag\"><mtext>(4.5)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Sei <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> eine Funktion und definiere <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">.<\/span> Gegeben <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi> <mi mathvariant=\"bold-script\">\u2131<\/mi> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> mit <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><\/math> kann man ebenso <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>u<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>u<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo> <\/mrow> <\/msub> <\/math> definieren. Es gilt <math display=\"inline\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> und <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Wegen Gleichung (<a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114002r4\">4.4<\/a>) erhalten wir, dass <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>u<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>u<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><msub><mrow><mi>u<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-114004r6\" \/><mstyle class=\"maketag\"><mtext>(4.6)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was wiederum <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> zur Folge hat. Umgekehrt kann man, gegeben Treppenfunktionen <math display=\"inline\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> mit <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> eine Treppenfunktion <math display=\"inline\"><mi>u<\/mi><\/math> auf <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> definieren, die ebenso <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><\/math> und Gleichung (<a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114004r6\">4.6<\/a>) erf\u00fcllt. <button class=\"hover-trigger\">(Wie genau?)<\/button><span class=\"hover-text\"><span class=\"marginpar\">Wegen Lemma <a href=\"..\/..\/chapter\/treppenfunktionen-und-deren-integral#x1-109004r5\">4.5<\/a> spielt der Funktionswert einer Treppenfunktion bei einem einzelnen Wert keine Rolle. Deswegen k\u00f6nnen wir beispielsweise die Funktion <math display=\"inline\"><mi>u<\/mi><\/math> definiert durch <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>u<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> verwenden.<\/span><\/span> Dadurch ist <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und wegen der Additionseigenschaft des Supremums in Proposition <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64008r63\">2.63<\/a> gilt <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-114005r7\" \/><mstyle class=\"maketag\"><mtext>(4.7)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Analog zeigt man, dass <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-114006r8\" \/><mstyle class=\"maketag\"><mtext>(4.8)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Ist nun <math display=\"inline\"><mi>f<\/mi><\/math> Riemann-integrierbar, dann ist <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2265<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2265<\/mo><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">\u00dcberall in dieser Kette von Ungleichungen gilt also Gleichheit. Somit ist <math display=\"inline\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">=<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi> <\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und dadurch auch <span class=\"maperiod\"><math display=\"inline\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi> <\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Das heisst, dass <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> und <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> Riemann-integrierbar sind und Gleichung&nbsp;(<a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114005r7\">4.7<\/a>) wird zur gew\u00fcnschten Additionseigenschaft f\u00fcr das Riemann-Integral. <\/p><p class=\"indent\">Falls <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> Riemann-integrierbar sind, dann gilt <math display=\"inline\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi> <\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Dies impliziert gemeinsam mit den Gleichungen&nbsp;(<a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114005r7\">4.7<\/a>), (<a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114006r8\">4.8<\/a>) auch <math display=\"inline\"><munder accentunder=\"false\" class=\"mml-underline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u0332<\/mo><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>I<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und die Additionseigenschaft. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Sei <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> ein kompaktes Intervall mit <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> Ist <math display=\"inline\"><mi>f<\/mi><\/math> eine Funktion, die auf einer gr\u00f6sseren Menge als <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> definiert ist, so werden wir anstelle von <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/msub><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> trotzdem meist <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> schreiben, wenn <math display=\"inline\"><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/msub><\/math> Riemann-integrierbar ist. Auch definieren wir die folgende Erweiterung des Riemann-Integrals <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>a<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace class=\"quad\" width=\"1em\" \/><mstyle class=\"text\"><mtext>und<\/mtext><\/mstyle><mspace class=\"quad\" width=\"1em\" \/><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>a<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-114007r9\" \/><mstyle class=\"maketag\"><mtext>(4.9)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Diese Definition vereinfacht die Notation und macht auf Grund der Aussage in folgender \u00dcbung Sinn. <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z91051fb2f94b\"> <a id=\"x1-114008r27\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.27 <\/span>(Intervalladditivit\u00e4t)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>I<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>b<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>b<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ein kompaktes Intervall und sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u211b<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>I<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie die Additionseigenschaft in Satz <\/span><a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114001r26\"><span class=\"ecti-1095\">4.26<\/span><\/a> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>I<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">L<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Wir unterscheiden F<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lle abh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngig von der Anordnung der Punkte<\/span> <math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>c<\/mi><\/math> <span class=\"ecti-1095\">im<\/span> <span class=\"ecti-1095\">Intervall <\/span><math display=\"inline\"><mi>I<\/mi><\/math> <span class=\"ecti-1095\">und zeigen jeweils, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-114009r10\" \/><mstyle class=\"maketag\"><mtext>(4.10)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">gilt wie gew<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nscht.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">1.<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Angenommen <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>c<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann ist<\/span> (<a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114009r10\">4.10<\/a>) <span class=\"ecti-1095\">genau die Additionseigenschaft aus Satz <\/span><a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114001r26\"><span class=\"ecti-1095\">4.26<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">2.<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Angenommen <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann ist <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">per Definition und es gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">3.<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>c<\/mi><\/math> <span class=\"ecti-1095\">gilt, so geht man wie in vorherigem Fall vor.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">4.<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Angenommen es gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>c<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann ist nach Satz <\/span><a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-114001r26\"><span class=\"ecti-1095\">4.26<\/span><\/a> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>a<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Somit gilt nach den Definitionen vor dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>a<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>b<\/mi><\/mrow><mrow><mi>c<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">5.<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die F<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>c<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>c<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>a<\/mi><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>a<\/mi><\/math> <span class=\"ecti-1095\">werden <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hnlich behandelt.<\/span><\/dd><\/dl> <div class=\"wp-nocaption \"><\/div><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z867f5c36b447\"> <a id=\"x1-114015r28\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.28 <\/span>(Stetigkeit des partikul\u00e4ren Integrals)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">und<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Riemann-integrierbare Funktion. Zeigen Sie, dass das sogenannte partikul<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re Integral<\/span> <\/p><table id=\"zb31a59c3a74f\" class=\"equation-star\"><tr><td> <math class=\"equation\" display=\"block\"> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> [<\/mo><mrow><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">]<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>x<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>t<\/mi> <\/math><\/td><\/tr><\/table> <p class=\"indent\"><span class=\"ecti-1095\">eine stetige reellwertige Funktion auf<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">definiert. Ist diese Funktion auch gleichm<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ssig oder Lipschitz-stetig (siehe <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/stetige-funktionen-auf-kompakten-intervallen#x1-102015r80\"><span class=\"ecti-1095\">3.80<\/span><\/a> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">letzteren Begriff)? <\/span><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"565px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/vvfnpyzd\/width\/688\/height\/565\/border\/888888\/rc\/false\/ai\/false\/sdz\/true\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><div class=\"wp-nocaption \"><\/div> <\/div> <a id=\"x1-114016r111\"><\/a> \n","protected":false},"author":1089,"menu_order":3,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-54","chapter","type-chapter","status-publish","hentry"],"part":51,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/54","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/users\/1089"}],"version-history":[{"count":0,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/54\/revisions"}],"part":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/parts\/51"}],"metadata":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/54\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=54"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=54"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=54"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=54"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}