{"id":53,"date":"2021-12-15T09:53:07","date_gmt":"2021-12-15T09:53:07","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/definition-des-riemann-integrals\/"},"modified":"2021-12-15T09:53:07","modified_gmt":"2021-12-15T09:53:07","slug":"definition-des-riemann-integrals","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/definition-des-riemann-integrals\/","title":{"raw":"Definition des Riemann-Integrals","rendered":"Definition des Riemann-Integrals"},"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: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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\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=\"z5bbe48ea184f\" class=\"sectionHead\"><span class=\"titlemark\">4.2 <\/span> <a id=\"x1-1100002\"><\/a>Definition des Riemann-Integrals<\/h3> <p class=\"noindent\">Wie schon im letzten Abschnitt betrachten wir im Folgenden Funktionen 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> zu reellen Zahlen <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Wir bemerken, dass Treppenfunktionen beschr\u00e4nkt sind, da sie endliche Bilder haben. Des Weiteren ist eine reellwertige Funktion <math display=\"inline\"><mi>f<\/mi><\/math> genau dann beschr\u00e4nkt, wenn es 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> gibt, die <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> erf\u00fcllen. In der Tat, falls&nbsp;<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> f\u00fcr gewisse Treppenfunktionen&nbsp;<math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>o<\/mi><\/math> gilt, dann ist <math display=\"inline\"><mi>f<\/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> von oben durch das Maximum von <math display=\"inline\"><mi>o<\/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> beschr\u00e4nkt und von unten durch das Minimum von <math display=\"inline\"><mi>u<\/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> beschr\u00e4nkt. (Wieso?). Umgekehrt k\u00f6nnen wir konstante 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> verwenden, falls <math display=\"inline\"><mi>f<\/mi><\/math> beschr\u00e4nkt ist. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z7b1aed6e1d41\"> <a id=\"x1-110001r10\"><\/a> <span class=\"ecbx-1095\">Definition 4.10.<\/span> <\/h4> <p class=\"indent\">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>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> beschr\u00e4nkt. Dann definieren wir die (nicht-leere) Menge der <span class=\"ecbx-1095\">Untersummen <\/span>durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <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>u<\/mi><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><mstyle class=\"text\"><mtext>&nbsp;und&nbsp;<\/mtext><\/mstyle><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <\/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\">und die (nicht-leere) Menge der <span class=\"ecbx-1095\">Obersummen <\/span>durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><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> <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>o<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>o<\/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><mstyle class=\"text\"><mtext>&nbsp;und&nbsp;<\/mtext><\/mstyle><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>o<\/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> <\/div> <p class=\"indent\">Falls ein \u201e vern\u00fcnftiges Integral\u201c <math display=\"inline\"><mi>I<\/mi><\/math> von <math display=\"inline\"><mi>f<\/mi><\/math> existiert, so sollte <math display=\"inline\"><mi>I<\/mi><\/math> eine obere Schranke von <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und eine untere Schranke von <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> sein. Wir wollen diese Beobachtung verwenden, um eine Definition des Integrals zu erarbeiten. <\/p><p class=\"indent\">F\u00fcr <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> wie in Definition <a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110001r10\">4.10<\/a> gilt nach Lemma <a href=\"..\/..\/chapter\/treppenfunktionen-und-deren-integral#x1-109009r8\">4.8<\/a> 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><mi>u<\/mi><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><mi>o<\/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\">Jede Untersumme ist also kleiner gleich jeder Obersumme. \u00c4quivalenterweise ist jede Obersumme <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>o<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>d<\/mi><mi>x<\/mi><\/math> eine obere Schranke der nicht-leeren Menge der Untersummen und daher ist                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/munderover><mi>o<\/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\">da das Supremum die kleinste obere Schranke ist. Insbesondere ist <math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> eine untere Schranke der Menge der Obersummen <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und es gilt <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo> <mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><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=\"x1-110002r2\" \/><mstyle class=\"maketag\"><mtext>(4.2)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">da das Infimum die gr\u00f6sste untere Schranke ist. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zc8de831c5fae\"> <a id=\"x1-110003r11\"><\/a> <span class=\"ecbx-1095\">Definition 4.11 <\/span>(Riemann-Integrierbarkeit)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">F\u00fcr eine beschr\u00e4nkte 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>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> wird <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><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> das <span class=\"ecbx-1095\">untere<\/span> <span class=\"ecbx-1095\">Integral <\/span>von <math display=\"inline\"><mi>f<\/mi><\/math> und <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><mi>f<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo> <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><\/math> das <span class=\"ecbx-1095\">obere Integral<\/span> von <math display=\"inline\"><mi>f<\/mi><\/math> genannt. Die Funktion <math display=\"inline\"><mi>f<\/mi><\/math> heisst <span class=\"ecbx-1095\">Riemann-integrierbar<\/span>, oder kurz <span class=\"ecbx-1095\">R-integrierbar<\/span>, falls                                                                                                                                                                           <span class=\"maperiod\"><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><\/span><span class=\"period\">.<\/span> In diesem Fall wird dieser gemeinsame Wert das <span class=\"ecbx-1095\">Riemann-Integral<\/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><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <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><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">genannt. Des Weiteren definieren wir<a id=\"dx1-110004\"><\/a> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi mathvariant=\"bold-script\">\u211b<\/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-rel\">=<\/mo><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">{<\/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 class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">}<\/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\">Wir bezeichnen <math display=\"inline\"><mi>a<\/mi><\/math> als die <span class=\"ecbx-1095\">untere <\/span>und <math display=\"inline\"><mi>b<\/mi><\/math> als die <span class=\"ecbx-1095\">obere Integrationsgrenze <\/span>und die Funktion als den <span class=\"ecbx-1095\">Integrand <\/span>f\u00fcr das Integral <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><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Wir haben hier den Zugang von Darboux f\u00fcr die Definition des Riemann-Integrals gew\u00e4hlt; in Kapitel <a href=\"..\/..\/part\/grenzwerte-reeller-folgen-und-funktionen#x1-1560006\">6<\/a> werden wir aber auch kurz die sogenannten Riemann-Summen besprechen, die von Riemann als Ausgangspunkt seiner Definition verwendet wurden. Es gibt neben diesen beiden \u00e4quivalenten Definitionen noch weitere, die wir nicht besprechen werden.                                                                                                                                                                           <\/p><p class=\"indent\">Falls <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> nicht-negativ (das heisst, es gilt <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math>), beschr\u00e4nkt und Riemann-integrierbar ist, dann interpretieren wir die Zahl <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> als den Fl\u00e4cheninhalt der Menge <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">{<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">}<\/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> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z8ca1dbbab69c\"> <a id=\"x1-110005r12\"><\/a> <span class=\"ecbx-1095\">Proposition 4.12 <\/span>(Charakterisierungen der Riemann-Integrierbarkeit)<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\">\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\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt. Folgende Bedingungen sind <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent:<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">ist Riemann-integrierbar.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es existiert h<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chstens eine (oder auch genau eine) reelle Zahl<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>I<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die die Ungleichungen<\/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>b<\/mi><\/mrow><\/msubsup><mi>u<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>I<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>o<\/mi><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\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><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> <span class=\"ecti-1095\">mit <\/span><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> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">existieren <\/span><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> <span class=\"ecti-1095\">mit <\/span><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><span class=\"ecti-1095\">, so<\/span> <span class=\"ecti-1095\">dass <\/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> <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><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><\/dd><\/dl> <\/div> <p class=\"indent\">Der dritte Punkt in obiger Proposition bedeutet intuitiv, dass <math display=\"inline\"><mi>f<\/mi><\/math> sich zwischen zwei Treppenfunktionen \u201eeinquetschen\u201c l\u00e4sst, so dass deren Differenz im Mittel (geometrisch formuliert, der Fl\u00e4cheninhalt zwischen den beiden Treppenfunktionen) klein 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\">Angenommen <math display=\"inline\"><mi>f<\/mi><\/math> ist Riemann-integrierbar wie in (i). Wir wollen (iii) zeigen. Sei also <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dann existiert (wegen der zweiten Charakterisierung des Supremums in Satz <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\">2.59<\/a>) ein <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>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><\/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><mi>u<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/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-bin\">\u2212<\/mo><mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> Genauso existiert ein <math display=\"inline\"><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>o<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>f<\/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><mi>o<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/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-bin\">+<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> Da <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> nach Voraussetzung folgt nun mit Lemma <a href=\"..\/..\/chapter\/treppenfunktionen-und-deren-integral#x1-109006r7\">4.7<\/a> <\/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><mi>o<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>u<\/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><mi>o<\/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>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>u<\/mi><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><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">&lt;<\/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-bin\">+<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><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-bin\">\u2212<\/mo><mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>\ud835\udf00<\/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\">wie in (iii) behauptet. <\/p><p class=\"indent\">Angenommen <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> ist beschr\u00e4nkt und erf\u00fcllt die Aussage in (iii). Wir wollen (ii) zeigen und nehmen also an, dass <math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> die Ungleichungen <\/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>u<\/mi><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\">\u2264<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>o<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>d<\/mi><mi>x<\/mi><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><mi>u<\/mi><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\">\u2264<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>o<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>d<\/mi><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\">f\u00fcr alle <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> erf\u00fcllen. F\u00fcr ein beliebiges <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> k\u00f6nnen wir wegen (iii) <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> finden, so dass die obigen Ungleichungen kombiniert zu                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>o<\/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>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>u<\/mi><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><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>o<\/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>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>u<\/mi><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><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fchren. Daher ist <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und es muss <math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> gelten. Dies zeigt, dass es h\u00f6chstens eine Zahl <math display=\"inline\"><mi>I<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> gibt, die die Ungleichung in (ii) erf\u00fcllt. <\/p><p class=\"indent\">Angenommen (ii) gilt. Wir behaupten, dass die Ungleichungen dann von genau einer Zahl erf\u00fcllt werden und dass <math display=\"inline\"><mi>f<\/mi><\/math> Riemann-integrierbar ist. In der Tat gilt nach Gleichung (<a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110002r2\">4.2<\/a>), dass <\/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>u<\/mi><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\">\u2264<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/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\">\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>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo> <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> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/munderover><mi>o<\/mi><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\">f\u00fcr alle <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 <span class=\"maperiod\"><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><\/span><span class=\"period\">.<\/span> Das heisst, dass sowohl <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><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> wie 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><\/math> die Ungleichungen in (ii) erf\u00fcllen. Nach Voraussetzung (von (ii)) folgt <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><mi>f<\/mi><\/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><\/math> und damit, dass <math display=\"inline\"><mi>f<\/mi><\/math> Riemann-integrierbar ist. <\/p><p class=\"indent\">Wir haben gesehen, dass die Implikationen (i)<math display=\"inline\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <\/math>(iii), (iii)<math display=\"inline\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <\/math>(ii) und (ii)<math display=\"inline\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <\/math>(i) gelten, also folgt die Proposition. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z87e845ee50f4\"> <a id=\"x1-110009r13\"><\/a> <span class=\"ecbx-1095\">Applet 4.13 <\/span>(Unter- und Obersummen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"425px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/amq7pc4b\/width\/688\/height\/425\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir sehen den Graph einer Funktion, k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen die betrachtete Zerlegung verfeinern (mit<\/span> <span class=\"ecti-1095\">dem Punkt <\/span><math display=\"inline\"><mo class=\"MathClass-bin\">+<\/mo><\/math><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">und dann (mit den Pfeilen) sowohl bessere Untersummen also auch besser Obersummen zu<\/span> <span class=\"ecti-1095\">der Funktion finden. K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen Sie die optimalen Unter- und Obersummen zu einer Zerlegung<\/span> <span class=\"ecti-1095\">in 5 Intervalle finden? Nach einigen Experimenten sollten Sie davon <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berzeugt sein, dass die<\/span> <span class=\"ecti-1095\">betrachtete Funktion Riemann-integrierbar ist \u2013 dies wird aus den sp<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">teren S<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzen dieses<\/span> <span class=\"ecti-1095\">Kapitels recht schnell folgen.<\/span> <\/p> <\/div> <p class=\"indent\">Gut zu wissen ist, dass das Riemann-Integral eine Verallgemeinerung des Integrals von Treppenfunktionen darstellt und in diesem Sinne auch einfach vom Riemann-Integral einer Treppenfunktion gesprochen werden kann. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z859cf00170ec\"> <a id=\"x1-110010r14\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.14 <\/span>(Zur Wohldefiniertheit)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/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\">eine                  Treppenfunktion.                  Zeigen                  Sie,                  dass<\/span> <math display=\"inline\"><mi>t<\/mi><\/math> <span class=\"ecti-1095\">Riemann-integrierbar        ist        und        dass        das        Riemann-Integral        von<\/span> <math display=\"inline\"><mi>t<\/mi><\/math> <span class=\"ecti-1095\">gleich                              dem                              Integral                              von<\/span> <math display=\"inline\"><mi>t<\/mi><\/math> <span class=\"ecti-1095\">als Treppenfunktion ist.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z7d66d2ab120c\"> <a id=\"x1-110011r15\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.15 <\/span>(Integral der Parabelfunktion)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wiederholen Sie den Beweis von Proposition<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/quadratur-der-parabel#x1-4004r1\"><span class=\"ecti-1095\">1.1<\/span><\/a> <span class=\"ecti-1095\">und zeigen Sie (in der Sprache dieses Abschnitts), dass<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">Riemann-integrierbar<\/span> <span class=\"ecti-1095\">ist mit <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Verifizieren Sie an dieser Stelle auch, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><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> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><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\">Die Charakterisierung (iii) in Proposition <a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110005r12\">4.12<\/a> ist unter anderem dann n\u00fctzlich, wenn man von spezifischen Funktionen die Riemann-Integrierbarkeit zeigen will. Ihre Bedingungen lassen sich sogar noch abschw\u00e4chen, was wir in folgender \u00dcbung diskutieren wollen. <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z36487348a83f\"> <a id=\"x1-110012r16\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.16 <\/span>(Betrachten spezieller Ober- und Untersummen)<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\">\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 beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkte<\/span> <span class=\"ecti-1095\">Funktion und sei <\/span><math display=\"inline\"><msub><mrow><mi>T<\/mi><\/mrow><mrow><mi>U<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine Menge<\/span> <span class=\"ecti-1095\">von Treppenfunktionen mit <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>T<\/mi><\/mrow><mrow><mi>U<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><msub><mrow><mi>T<\/mi><\/mrow><mrow><mi>O<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">eine Menge von<\/span> <span class=\"ecti-1095\">Treppenfunktionen mit <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>o<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>o<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>T<\/mi><\/mrow><mrow><mi>O<\/mi><\/mrow><\/msub><\/math><span class=\"ecti-1095\">. Angenommen<\/span> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">existieren <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>T<\/mi><\/mrow><mrow><mi>U<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>o<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>T<\/mi><\/mrow><mrow><mi>O<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">mit<\/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><mi>o<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>u<\/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\"><span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">Riemann-integrierbar ist 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><mi>f<\/mi><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> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/munderover><mi>u<\/mi><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> <msub><mrow><mi>T<\/mi><\/mrow><mrow> <mi>U<\/mi><\/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> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/munderover><mi>o<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>o<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>T<\/mi><\/mrow><mrow> <mi>O<\/mi><\/mrow><\/msub><\/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\"><\/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 den Beweis von Proposition <\/span><a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110005r12\"><span class=\"ecti-1095\">4.12<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z309addc11fc8\"> <a id=\"x1-110013r17\"><\/a> <span class=\"ecbx-1095\">Beispiel 4.17 <\/span>(Eine nicht-Riemann-integrierbare Funktion)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten wieder die sogenannte Dirichlet-Funktion, das heisst, die charakteristische<\/span> <span class=\"ecti-1095\">Funktion<\/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> <msub><mrow><mi>\ud835\udfd9<\/mi><\/mrow> <mrow> <mi>\u211a<\/mi><mo class=\"MathClass-bin\">\u2229<\/mo><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/msub> <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=\"center\"><mn>1<\/mn><\/mtd><mtd class=\"array\" columnalign=\"left\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211a<\/mi><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>0<\/mn><\/mtd> <mtd class=\"array\" columnalign=\"left\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u2209<\/mo> <mi>\u211a<\/mi><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/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\"><span class=\"ecti-1095\">Die Behauptung ist, dass diese nicht Riemann-integrierbar ist. Dazu berechnen wir das untere und das obere<\/span> <span class=\"ecti-1095\">Integral von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><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><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">mit<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>o<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sei<\/span> <\/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><mn>0<\/mn> <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> <mn>1<\/mn><\/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\">eine Zerlegung in Konstanzintervalle von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>o<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sei <\/span><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> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <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 class=\"ecti-1095\">. Da<\/span> <math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">dicht in<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist (siehe Korollar<\/span> <a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-68006r70\"><span class=\"ecti-1095\">2.70<\/span><\/a><span class=\"ecti-1095\">), existiert ein <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <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 class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211a<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Wegen <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>o<\/mi><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>1<\/mn> <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\">\u2264<\/mo> <mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Somit gilt<\/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><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/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><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> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2265<\/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> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/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\">unter Verwendung von Teleskopsummen. Damit ist das obere Integral von<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">durch<\/span> <math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">gegeben, da die Treppenfunktion<\/span> <span class=\"ecti-1095\">mit konstantem Wert <\/span><math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">Integral <\/span><math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">hat und <\/span><math display=\"inline\"><mi>o<\/mi><\/math> <span class=\"ecti-1095\">beliebig war. <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">hnlich (siehe <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110014r18\"><span class=\"ecti-1095\">4.18<\/span><\/a><span class=\"ecti-1095\">) zeigt man, dass das untere Integral von<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">durch<\/span> <math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">gegeben ist.<\/span> <span class=\"ecti-1095\">Somit ist <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">nicht Riemann-integrierbar.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Es ist etwas schwierig den Graphen der Dirichlet-Funktion zu zeichnen (vor allem da f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die<\/span> <span class=\"ecti-1095\">meisten Computerprogramme alle Zahlen rational sind). Wir wollen dies aber trotzdem<\/span> <span class=\"ecti-1095\">versuchen, wobei die verschiedenen Kreuze die Funktionswerte der ersten rationalen Zahlen<\/span> <span class=\"ecti-1095\">andeuten.<\/span> <\/p><p class=\"indent\"><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"347px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/Kxq4rg9V\/width\/688\/height\/347\/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> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z5a357ad2ef95\"> <a id=\"x1-110014r18\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.18.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen                   Sie,                    dass                   die                   Funktion<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">aus                   Beispiel                   <\/span><a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110013r17\"><span class=\"ecti-1095\">4.17<\/span><\/a> <span class=\"ecti-1095\">unteres                   Integral<\/span> <math display=\"inline\"><mn>0<\/mn><\/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\">Gehen   Sie   genauso   wie   im   Beispiel   vor   und   zeigen   Sie   dazu,   dass<\/span> <math display=\"inline\"><mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">dicht                                                                                                          in<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">liegt. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r       letzteres       kann       man       beispielsweise       Dichtheit       von<\/span> <math display=\"inline\"><msqrt><mrow> <mn>2<\/mn><\/mrow><\/msqrt> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">zeigen.<\/span><\/p><\/details>  <\/div> <a id=\"x1-110015r110\"><\/a> \n","rendered":"\n<style scoped=\"scoped\">.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: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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width:125%;}\ndt {text-align:right; font-weight:bold; clear:left; float:left;}\ndd {width:100%; padding-left:1em; padding-top: 0px; clear:right;}\ndd + dd {float:right; clear:both;}\ndd + dt {clear:both;}\ndt + dt {width: 100%; float: none; padding: 0 70% 0 0;}\ndt + dt + dd {margin-top: -2em;}\ndt + dt + dd + dt {margin-top: 2em;}\n<\/style>\n<style scoped=\"scoped\">\n\/* CSS Analysis-Skript D-Math ETHZ *\/\n\n\/* Uniform Font, also for headers *\/\nh3 {\n\tfont-family: \"Times New Roman\", serif;\n\tmargin-bottom: 35px;\n}\nh4 {\n\tfont-family: \"Times New Roman\", serif;\n}\nh5 {\n\tfont-family: \"Times New Roman\", serif;\n}\n\n\/* Bold font, e.g. for definitions *\/\n.ecbx-1095 {font-weight: 550 ;}\n\n\n\/* Uniform spacing, indent: larger, noindent, enumerate, itemize *\/\np.indent {\n\tmargin: 25px 0px 0px 0px;\n\ttext-indent: 0px; \n}\np.noindent {\n\tmargin: 15px 0px 0px 0px;\n\ttext-indent: 0px; \n}\ndl.enumerate {\n\tmargin: 0px 0px 0px 0px;\n}\ndl.enumerate dt, dl.enumerate dd {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\ndiv.custom-itemize {\n\tmargin: 0px 0px 0px 0px;\n}\ndiv.custom-itemize div.item-head {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n\ttext-align: center;\n}\ndiv.custom-itemize div.item-head:first-of-type {\n\tmargin-top: 0px;\n} \ndiv.custom-itemize div.item-content {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\n.MJXc-display {\n\tmargin: 15px 0px 0px 0px;\n}\n\n\n\n\/* green metheorem\/melemma CSS class for more\/medium important latex-theorem-environments *\/\n\/* metheorem box+header *\/\ndiv.metheorem {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #c7e4da;\n}\ndiv.metheorem h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* melemma box+header *\/\ndiv.melemma {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #F2F2F2;\n}\ndiv.melemma h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* meexample box+header *\/\ndiv.meexample {\n    margin-bottom: 30px;\n    margin-top: 30px;\n\tpadding: 0px 15px 15px 15px;\n\tborder-color: gainsboro;\n\tborder-style: solid;\n\tborder-width: thin;\n}\ndiv.meexample h4 {\n\tfont-size: inherit;\n\tfont-weight: bold;\n    padding: 15px 0px 0px 0px;\n\tmargin-top: 0px;\n\tmargin-bottom: 5px;\n}\ndiv.meexample h4+p.noindent, div.meexample h4+p.indent {\n\tmargin-top: 5px;\n\ttext-indent: 0px;\n}\n\/* padding and margins for stuff inside these boxes, CSS-selector &gt; doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                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=\"z5bbe48ea184f\" class=\"sectionHead\"><span class=\"titlemark\">4.2 <\/span> <a id=\"x1-1100002\"><\/a>Definition des Riemann-Integrals<\/h3> <p class=\"noindent\">Wie schon im letzten Abschnitt betrachten wir im Folgenden Funktionen 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> zu reellen Zahlen <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Wir bemerken, dass Treppenfunktionen beschr\u00e4nkt sind, da sie endliche Bilder haben. Des Weiteren ist eine reellwertige Funktion <math display=\"inline\"><mi>f<\/mi><\/math> genau dann beschr\u00e4nkt, wenn es 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> gibt, die <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> erf\u00fcllen. In der Tat, falls&nbsp;<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> f\u00fcr gewisse Treppenfunktionen&nbsp;<math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>o<\/mi><\/math> gilt, dann ist <math display=\"inline\"><mi>f<\/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> von oben durch das Maximum von <math display=\"inline\"><mi>o<\/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> beschr\u00e4nkt und von unten durch das Minimum von <math display=\"inline\"><mi>u<\/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> beschr\u00e4nkt. (Wieso?). Umgekehrt k\u00f6nnen wir konstante 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> verwenden, falls <math display=\"inline\"><mi>f<\/mi><\/math> beschr\u00e4nkt ist. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z7b1aed6e1d41\"> <a id=\"x1-110001r10\"><\/a> <span class=\"ecbx-1095\">Definition 4.10.<\/span> <\/h4> <p class=\"indent\">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>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> beschr\u00e4nkt. Dann definieren wir die (nicht-leere) Menge der <span class=\"ecbx-1095\">Untersummen <\/span>durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <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>u<\/mi><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><mstyle class=\"text\"><mtext>&nbsp;und&nbsp;<\/mtext><\/mstyle><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <\/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\">und die (nicht-leere) Menge der <span class=\"ecbx-1095\">Obersummen <\/span>durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><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> <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>o<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>o<\/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><mstyle class=\"text\"><mtext>&nbsp;und&nbsp;<\/mtext><\/mstyle><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>o<\/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> <\/div> <p class=\"indent\">Falls ein \u201e vern\u00fcnftiges Integral\u201c <math display=\"inline\"><mi>I<\/mi><\/math> von <math display=\"inline\"><mi>f<\/mi><\/math> existiert, so sollte <math display=\"inline\"><mi>I<\/mi><\/math> eine obere Schranke von <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und eine untere Schranke von <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> sein. Wir wollen diese Beobachtung verwenden, um eine Definition des Integrals zu erarbeiten. <\/p><p class=\"indent\">F\u00fcr <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> wie in Definition <a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110001r10\">4.10<\/a> gilt nach Lemma <a href=\"..\/..\/chapter\/treppenfunktionen-und-deren-integral#x1-109009r8\">4.8<\/a> 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><mi>u<\/mi><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><mi>o<\/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\">Jede Untersumme ist also kleiner gleich jeder Obersumme. \u00c4quivalenterweise ist jede Obersumme <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>o<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>d<\/mi><mi>x<\/mi><\/math> eine obere Schranke der nicht-leeren Menge der Untersummen und daher ist                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/munderover><mi>o<\/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\">da das Supremum die kleinste obere Schranke ist. Insbesondere ist <math display=\"inline\"><mi class=\"qopname\">sup<\/mi><mo>  <\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> eine untere Schranke der Menge der Obersummen <math display=\"inline\"><mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und es gilt <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo> <mi mathvariant=\"bold-script\">\ud835\udcaa<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><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=\"x1-110002r2\" \/><mstyle class=\"maketag\"><mtext>(4.2)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">da das Infimum die gr\u00f6sste untere Schranke ist. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zc8de831c5fae\"> <a id=\"x1-110003r11\"><\/a> <span class=\"ecbx-1095\">Definition 4.11 <\/span>(Riemann-Integrierbarkeit)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">F\u00fcr eine beschr\u00e4nkte 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>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> wird <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><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> das <span class=\"ecbx-1095\">untere<\/span> <span class=\"ecbx-1095\">Integral <\/span>von <math display=\"inline\"><mi>f<\/mi><\/math> und <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><mi>f<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo> <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><\/math> das <span class=\"ecbx-1095\">obere Integral<\/span> von <math display=\"inline\"><mi>f<\/mi><\/math> genannt. Die Funktion <math display=\"inline\"><mi>f<\/mi><\/math> heisst <span class=\"ecbx-1095\">Riemann-integrierbar<\/span>, oder kurz <span class=\"ecbx-1095\">R-integrierbar<\/span>, falls                                                                                                                                                                           <span class=\"maperiod\"><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><\/span><span class=\"period\">.<\/span> In diesem Fall wird dieser gemeinsame Wert das <span class=\"ecbx-1095\">Riemann-Integral<\/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><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <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><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">genannt. Des Weiteren definieren wir<a id=\"dx1-110004\"><\/a> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi mathvariant=\"bold-script\">\u211b<\/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-rel\">=<\/mo><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">{<\/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 class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">}<\/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\">Wir bezeichnen <math display=\"inline\"><mi>a<\/mi><\/math> als die <span class=\"ecbx-1095\">untere <\/span>und <math display=\"inline\"><mi>b<\/mi><\/math> als die <span class=\"ecbx-1095\">obere Integrationsgrenze <\/span>und die Funktion als den <span class=\"ecbx-1095\">Integrand <\/span>f\u00fcr das Integral <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><mi>f<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Wir haben hier den Zugang von Darboux f\u00fcr die Definition des Riemann-Integrals gew\u00e4hlt; in Kapitel <a href=\"..\/..\/part\/grenzwerte-reeller-folgen-und-funktionen#x1-1560006\">6<\/a> werden wir aber auch kurz die sogenannten Riemann-Summen besprechen, die von Riemann als Ausgangspunkt seiner Definition verwendet wurden. Es gibt neben diesen beiden \u00e4quivalenten Definitionen noch weitere, die wir nicht besprechen werden.                                                                                                                                                                           <\/p><p class=\"indent\">Falls <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> nicht-negativ (das heisst, es gilt <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math>), beschr\u00e4nkt und Riemann-integrierbar ist, dann interpretieren wir die Zahl <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> als den Fl\u00e4cheninhalt der Menge <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">{<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2223<\/mo><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mn>0<\/mn> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">}<\/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> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z8ca1dbbab69c\"> <a id=\"x1-110005r12\"><\/a> <span class=\"ecbx-1095\">Proposition 4.12 <\/span>(Charakterisierungen der Riemann-Integrierbarkeit)<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\">\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\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt. Folgende Bedingungen sind <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent:<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">ist Riemann-integrierbar.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Es existiert h<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chstens eine (oder auch genau eine) reelle Zahl<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>I<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">die die Ungleichungen<\/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>b<\/mi><\/mrow><\/msubsup><mi>u<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>I<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>o<\/mi><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\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><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> <span class=\"ecti-1095\">mit <\/span><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> <span class=\"ecti-1095\">erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">existieren <\/span><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> <span class=\"ecti-1095\">mit <\/span><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><span class=\"ecti-1095\">, so<\/span> <span class=\"ecti-1095\">dass <\/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> <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><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><\/dd><\/dl> <\/div> <p class=\"indent\">Der dritte Punkt in obiger Proposition bedeutet intuitiv, dass <math display=\"inline\"><mi>f<\/mi><\/math> sich zwischen zwei Treppenfunktionen \u201eeinquetschen\u201c l\u00e4sst, so dass deren Differenz im Mittel (geometrisch formuliert, der Fl\u00e4cheninhalt zwischen den beiden Treppenfunktionen) klein 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\">Angenommen <math display=\"inline\"><mi>f<\/mi><\/math> ist Riemann-integrierbar wie in (i). Wir wollen (iii) zeigen. Sei also <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Dann existiert (wegen der zweiten Charakterisierung des Supremums in Satz <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\">2.59<\/a>) ein <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>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><\/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><mi>u<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/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-bin\">\u2212<\/mo><mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> Genauso existiert ein <math display=\"inline\"><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>o<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>f<\/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><mi>o<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/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-bin\">+<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> Da <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> nach Voraussetzung folgt nun mit Lemma <a href=\"..\/..\/chapter\/treppenfunktionen-und-deren-integral#x1-109006r7\">4.7<\/a> <\/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><mi>o<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>u<\/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><mi>o<\/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>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>u<\/mi><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><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">&lt;<\/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-bin\">+<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><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-bin\">\u2212<\/mo><mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>\ud835\udf00<\/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\">wie in (iii) behauptet. <\/p><p class=\"indent\">Angenommen <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> ist beschr\u00e4nkt und erf\u00fcllt die Aussage in (iii). Wir wollen (ii) zeigen und nehmen also an, dass <math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> die Ungleichungen <\/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>u<\/mi><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\">\u2264<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>o<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>d<\/mi><mi>x<\/mi><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><mi>u<\/mi><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\">\u2264<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>o<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>d<\/mi><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\">f\u00fcr alle <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> erf\u00fcllen. F\u00fcr ein beliebiges <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> k\u00f6nnen wir wegen (iii) <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> finden, so dass die obigen Ungleichungen kombiniert zu                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>o<\/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>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>u<\/mi><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><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>o<\/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>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>u<\/mi><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><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fchren. Daher ist <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und es muss <math display=\"inline\"><msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>I<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> gelten. Dies zeigt, dass es h\u00f6chstens eine Zahl <math display=\"inline\"><mi>I<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> gibt, die die Ungleichung in (ii) erf\u00fcllt. <\/p><p class=\"indent\">Angenommen (ii) gilt. Wir behaupten, dass die Ungleichungen dann von genau einer Zahl erf\u00fcllt werden und dass <math display=\"inline\"><mi>f<\/mi><\/math> Riemann-integrierbar ist. In der Tat gilt nach Gleichung (<a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110002r2\">4.2<\/a>), dass <\/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>u<\/mi><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\">\u2264<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/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\">\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>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo> <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> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/munderover><mi>o<\/mi><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\">f\u00fcr alle <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 <span class=\"maperiod\"><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><\/span><span class=\"period\">.<\/span> Das heisst, dass sowohl <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><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> wie 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><\/math> die Ungleichungen in (ii) erf\u00fcllen. Nach Voraussetzung (von (ii)) folgt <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><mi>f<\/mi><\/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><\/math> und damit, dass <math display=\"inline\"><mi>f<\/mi><\/math> Riemann-integrierbar ist. <\/p><p class=\"indent\">Wir haben gesehen, dass die Implikationen (i)<math display=\"inline\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <\/math>(iii), (iii)<math display=\"inline\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <\/math>(ii) und (ii)<math display=\"inline\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <\/math>(i) gelten, also folgt die Proposition. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z87e845ee50f4\"> <a id=\"x1-110009r13\"><\/a> <span class=\"ecbx-1095\">Applet 4.13 <\/span>(Unter- und Obersummen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <div class=\"wp-nocaption \"><\/div><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"425px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/amq7pc4b\/width\/688\/height\/425\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir sehen den Graph einer Funktion, k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen die betrachtete Zerlegung verfeinern (mit<\/span> <span class=\"ecti-1095\">dem Punkt <\/span><math display=\"inline\"><mo class=\"MathClass-bin\">+<\/mo><\/math><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">und dann (mit den Pfeilen) sowohl bessere Untersummen also auch besser Obersummen zu<\/span> <span class=\"ecti-1095\">der Funktion finden. K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen Sie die optimalen Unter- und Obersummen zu einer Zerlegung<\/span> <span class=\"ecti-1095\">in 5 Intervalle finden? Nach einigen Experimenten sollten Sie davon <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berzeugt sein, dass die<\/span> <span class=\"ecti-1095\">betrachtete Funktion Riemann-integrierbar ist \u2013 dies wird aus den sp<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">teren S<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tzen dieses<\/span> <span class=\"ecti-1095\">Kapitels recht schnell folgen.<\/span> <\/p> <\/div> <p class=\"indent\">Gut zu wissen ist, dass das Riemann-Integral eine Verallgemeinerung des Integrals von Treppenfunktionen darstellt und in diesem Sinne auch einfach vom Riemann-Integral einer Treppenfunktion gesprochen werden kann. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z859cf00170ec\"> <a id=\"x1-110010r14\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.14 <\/span>(Zur Wohldefiniertheit)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/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\">eine                  Treppenfunktion.                  Zeigen                  Sie,                  dass<\/span> <math display=\"inline\"><mi>t<\/mi><\/math> <span class=\"ecti-1095\">Riemann-integrierbar        ist        und        dass        das        Riemann-Integral        von<\/span> <math display=\"inline\"><mi>t<\/mi><\/math> <span class=\"ecti-1095\">gleich                              dem                              Integral                              von<\/span> <math display=\"inline\"><mi>t<\/mi><\/math> <span class=\"ecti-1095\">als Treppenfunktion ist.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z7d66d2ab120c\"> <a id=\"x1-110011r15\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.15 <\/span>(Integral der Parabelfunktion)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wiederholen Sie den Beweis von Proposition<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/quadratur-der-parabel#x1-4004r1\"><span class=\"ecti-1095\">1.1<\/span><\/a> <span class=\"ecti-1095\">und zeigen Sie (in der Sprache dieses Abschnitts), dass<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">Riemann-integrierbar<\/span> <span class=\"ecti-1095\">ist mit <\/span><span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msubsup><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Verifizieren Sie an dieser Stelle auch, dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi mathvariant=\"bold-script\">\ud835\udcb0<\/mi><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-punc\">,<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><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> <mo class=\"MathClass-rel\">=<\/mo> <mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>3<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><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\">Die Charakterisierung (iii) in Proposition <a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110005r12\">4.12<\/a> ist unter anderem dann n\u00fctzlich, wenn man von spezifischen Funktionen die Riemann-Integrierbarkeit zeigen will. Ihre Bedingungen lassen sich sogar noch abschw\u00e4chen, was wir in folgender \u00dcbung diskutieren wollen. <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z36487348a83f\"> <a id=\"x1-110012r16\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.16 <\/span>(Betrachten spezieller Ober- und Untersummen)<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\">\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 beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkte<\/span> <span class=\"ecti-1095\">Funktion und sei <\/span><math display=\"inline\"><msub><mrow><mi>T<\/mi><\/mrow><mrow><mi>U<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">eine Menge<\/span> <span class=\"ecti-1095\">von Treppenfunktionen mit <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>T<\/mi><\/mrow><mrow><mi>U<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><msub><mrow><mi>T<\/mi><\/mrow><mrow><mi>O<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">eine Menge von<\/span> <span class=\"ecti-1095\">Treppenfunktionen mit <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>o<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>o<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>T<\/mi><\/mrow><mrow><mi>O<\/mi><\/mrow><\/msub><\/math><span class=\"ecti-1095\">. Angenommen<\/span> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">existieren <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>T<\/mi><\/mrow><mrow><mi>U<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>o<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>T<\/mi><\/mrow><mrow><mi>O<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">mit<\/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><mi>o<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>u<\/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\"><span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">Riemann-integrierbar ist 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><mi>f<\/mi><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> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/munderover><mi>u<\/mi><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> <msub><mrow><mi>T<\/mi><\/mrow><mrow> <mi>U<\/mi><\/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> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/munderover><mi>o<\/mi><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>o<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>T<\/mi><\/mrow><mrow> <mi>O<\/mi><\/mrow><\/msub><\/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> <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 den Beweis von Proposition <\/span><a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110005r12\"><span class=\"ecti-1095\">4.12<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z309addc11fc8\"> <a id=\"x1-110013r17\"><\/a> <span class=\"ecbx-1095\">Beispiel 4.17 <\/span>(Eine nicht-Riemann-integrierbare Funktion)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten wieder die sogenannte Dirichlet-Funktion, das heisst, die charakteristische<\/span> <span class=\"ecti-1095\">Funktion<\/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> <msub><mrow><mi>\ud835\udfd9<\/mi><\/mrow> <mrow> <mi>\u211a<\/mi><mo class=\"MathClass-bin\">\u2229<\/mo><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/mrow><\/msub> <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=\"center\"><mn>1<\/mn><\/mtd><mtd class=\"array\" columnalign=\"left\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211a<\/mi><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mn>0<\/mn><\/mtd> <mtd class=\"array\" columnalign=\"left\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u2209<\/mo> <mi>\u211a<\/mi><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/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\"><span class=\"ecti-1095\">Die Behauptung ist, dass diese nicht Riemann-integrierbar ist. Dazu berechnen wir das untere und das obere<\/span> <span class=\"ecti-1095\">Integral von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><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><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">mit<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>o<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sei<\/span> <\/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><mn>0<\/mn> <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> <mn>1<\/mn><\/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\">eine Zerlegung in Konstanzintervalle von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>o<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Sei <\/span><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> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <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 class=\"ecti-1095\">. Da<\/span> <math display=\"inline\"><mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">dicht in<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist (siehe Korollar<\/span> <a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-68006r70\"><span class=\"ecti-1095\">2.70<\/span><\/a><span class=\"ecti-1095\">), existiert ein <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <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 class=\"ecti-1095\">mit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211a<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Wegen <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>o<\/mi><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>1<\/mn> <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\">\u2264<\/mo> <mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Somit gilt<\/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><mn>0<\/mn><\/mrow><mrow><mn>1<\/mn><\/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><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> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2265<\/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> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/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\">unter Verwendung von Teleskopsummen. Damit ist das obere Integral von<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">durch<\/span> <math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">gegeben, da die Treppenfunktion<\/span> <span class=\"ecti-1095\">mit konstantem Wert <\/span><math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">Integral <\/span><math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">hat und <\/span><math display=\"inline\"><mi>o<\/mi><\/math> <span class=\"ecti-1095\">beliebig war. <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">hnlich (siehe <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110014r18\"><span class=\"ecti-1095\">4.18<\/span><\/a><span class=\"ecti-1095\">) zeigt man, dass das untere Integral von<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">durch<\/span> <math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">gegeben ist.<\/span> <span class=\"ecti-1095\">Somit ist <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">nicht Riemann-integrierbar.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Es ist etwas schwierig den Graphen der Dirichlet-Funktion zu zeichnen (vor allem da f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die<\/span> <span class=\"ecti-1095\">meisten Computerprogramme alle Zahlen rational sind). Wir wollen dies aber trotzdem<\/span> <span class=\"ecti-1095\">versuchen, wobei die verschiedenen Kreuze die Funktionswerte der ersten rationalen Zahlen<\/span> <span class=\"ecti-1095\">andeuten.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"347px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/Kxq4rg9V\/width\/688\/height\/347\/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> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z5a357ad2ef95\"> <a id=\"x1-110014r18\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 4.18.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen                   Sie,                    dass                   die                   Funktion<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">aus                   Beispiel                   <\/span><a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110013r17\"><span class=\"ecti-1095\">4.17<\/span><\/a> <span class=\"ecti-1095\">unteres                   Integral<\/span> <math display=\"inline\"><mn>0<\/mn><\/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\">Gehen   Sie   genauso   wie   im   Beispiel   vor   und   zeigen   Sie   dazu,   dass<\/span> <math display=\"inline\"><mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">dicht                                                                                                          in<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">liegt. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r       letzteres       kann       man       beispielsweise       Dichtheit       von<\/span> <math display=\"inline\"><msqrt><mrow> <mn>2<\/mn><\/mrow><\/msqrt> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u211a<\/mi><\/math> <span class=\"ecti-1095\">zeigen.<\/span><\/p><\/details>  <\/div> <a id=\"x1-110015r110\"><\/a> \n","protected":false},"author":1089,"menu_order":2,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-53","chapter","type-chapter","status-publish","hentry"],"part":51,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/53","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\/53\/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\/53\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=53"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=53"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=53"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=53"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}