{"id":58,"date":"2021-12-15T09:53:08","date_gmt":"2021-12-15T09:53:08","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/integrierbarkeit-stetiger-funktionen\/"},"modified":"2021-12-15T09:53:08","modified_gmt":"2021-12-15T09:53:08","slug":"integrierbarkeit-stetiger-funktionen","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/integrierbarkeit-stetiger-funktionen\/","title":{"raw":"Integrierbarkeit stetiger Funktionen","rendered":"Integrierbarkeit stetiger Funktionen"},"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=\"zd64c49abb2e0\" class=\"sectionHead\"><span class=\"titlemark\">4.7 <\/span> <a id=\"x1-1230007\"><\/a>Integrierbarkeit stetiger Funktionen<\/h3> <p class=\"noindent\">Aus Abschnitt <a href=\"..\/..\/chapter\/integration-von-polynomen#x1-1220006\">4.6<\/a> wissen wir bereits, dass Polynomfunktionen integrierbar sind. In diesem Abschnitt m\u00f6chten wir nun unter Verwendung der Beschr\u00e4nktheit und der gleichm\u00e4ssigen Stetigkeit stetiger Funktionen auf kompakten Intervallen (Satz <a href=\"..\/..\/chapter\/stetige-funktionen-auf-kompakten-intervallen#x1-100001r69\">3.69<\/a> respektive Satz <a href=\"..\/..\/chapter\/stetige-funktionen-auf-kompakten-intervallen#x1-102005r77\">3.77<\/a>) folgendes allgemeines Resultat beweisen. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z4c4e159b10fa\"> <a id=\"x1-123001r42\"><\/a> <span class=\"ecbx-1095\">Satz 4.42 <\/span>(Stetige Funktionen und das Riemann-Integral)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Eine stetige Funktion auf einem kompakten Intervall <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">ist Riemann-integrierbar.<\/span> <\/p> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Sei <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>C<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Nach Satz <a href=\"..\/..\/chapter\/stetige-funktionen-auf-kompakten-intervallen#x1-102005r77\">3.77<\/a> ist <math display=\"inline\"><mi>f<\/mi><\/math> gleichm\u00e4ssig stetig und es gibt ein <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> so dass f\u00fcr alle <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> gilt <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <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\"><mstyle class=\"label\" id=\"x1-123002r16\" \/><mstyle class=\"maketag\"><mtext>(4.16)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Sei <math display=\"inline\"><mi>\u2128<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> eine Zerlegung von <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">\u2026<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><\/munder> <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\">&lt;<\/mo> <mi>\u03b4<\/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\">Zum Beispiel k\u00f6nnen wir die Zerlegung durch <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>k<\/mi> <mfrac> <mrow> <mi>b<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>a<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/math> f\u00fcr <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi><\/math> und ein hinreichend grosses <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> definieren. Wir definieren f\u00fcr jedes <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> die Zahlen <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> min<\/mi><mo>  <\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/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><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/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><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wobei wir Korollar <a href=\"..\/..\/chapter\/stetige-funktionen-auf-kompakten-intervallen#x1-101001r71\">3.71<\/a> f\u00fcr die Existenz von Minimum und Maximum verwendet haben. Wir behaupten nun, dass f\u00fcr alle <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> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <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\">gilt. In der Tat ist <math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi class=\"qopname\">min<\/mi><mo>  <\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi class=\"qopname\">min<\/mi><mo>  <\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo> <\/mrow> <\/msub> <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><span class=\"period\">.<\/span> Da aber <math display=\"inline\"><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> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><\/math> ist, haben wir auch <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi class=\"qopname\">min<\/mi><mo>  <\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><\/math><\/span><span class=\"period\">.<\/span> Auf Grund der Wahl von <math display=\"inline\"><mi>\u03b4<\/mi><\/math> mit (<a href=\"..\/..\/chapter\/integrierbarkeit-stetiger-funktionen#x1-123002r16\">4.16<\/a>) erhalten wir also unsere Behauptung <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi class=\"qopname\">min<\/mi><mo>  <\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Wir definieren nun Treppenfunktionen <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>o<\/mi><\/math> durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><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><mstyle class=\"text\"><mtext>&nbsp;f\u00fcr&nbsp;<\/mtext><\/mstyle><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><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi> <\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/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\"><mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><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><mstyle class=\"text\"><mtext>&nbsp;f\u00fcr&nbsp;<\/mtext><\/mstyle><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><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi> <\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> Nach Definition von <math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> f\u00fcr <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> gilt daher <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> Des Weiteren ist                                                                                                                                                                           <\/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\">=<\/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>M<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <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\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><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> <mi>\ud835\udf00<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>b<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Da <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig war (und <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/math> fix ist), zeigt dies mittels Proposition <a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110005r12\">4.12<\/a>, dass <math display=\"inline\"><mi>f<\/mi><\/math> Riemann-integrierbar ist. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z8a6bf1f311ec\"> <a id=\"x1-123003r43\"><\/a> <span class=\"ecbx-1095\">Applet 4.43 <\/span>(Integrierbarkeit einer \u201ezittrigen\u201c Funktion)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"652px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/ms4mjkun\/width\/688\/height\/652\/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, dass eine stetige aber zittrige Funktion wie im dargestellten Graphen auch<\/span> <span class=\"ecti-1095\">Riemann-integrierbar ist.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Anmerkung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Was wir auch mitunter sehen k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen, ist, dass geogebra mit der verwendeten<\/span> <span class=\"ecti-1095\">Funktion manchmal  Problem  hat  und  manche  der  dargestellten  Untersummen  oder<\/span> <span class=\"ecti-1095\">Obersummen eigentlich nicht richtig dargestellt und berechnet werden. Unabh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngig davon<\/span> <span class=\"ecti-1095\">haben wir aber in unserem Beweis schon die Riemann-Integrierbarkeit gesehen, sind also f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">die gew<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nschte Aussage nicht auf geogebra angewiesen.<\/span><\/p><\/details>  <\/div> <a id=\"x1-123004r120\"><\/a> <h4 id=\"z5e8a27086cff\" class=\"subsectionHead\"><span class=\"titlemark\">4.7.1 <\/span> <a id=\"x1-1240001\"><\/a>Sandwich mittels stetigen Funktionen<\/h4> <p class=\"noindent\">Die folgende Proposition zeigt, dass auch allgemeine Riemann-integrierbare Funktionen einen gewissen Zusammenhang zu dem Begriff der Stetigkeit aufweisen. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z778781f18003\"> <a id=\"x1-124001r44\"><\/a> <span class=\"ecbx-1095\">Proposition 4.44 <\/span>(Sandwich-Kriterium mit stetigen Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Eine Funktion <\/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\">auf<\/span> <span class=\"ecti-1095\">einem kompakten Intervall <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">ist genau dann<\/span> <span class=\"ecti-1095\">Riemann-integrierbar, wenn 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\">stetige Funktionen <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">existieren mit <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/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> <\/div> <p class=\"indent\">Wir empfehlen Ihnen sich dies geometrisch vorzustellen: <math display=\"inline\"><mi>f<\/mi><\/math> wird von den stetigen Funktionen <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/math> und <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <\/math> eingez\u00e4unt, wobei die Fl\u00e4che zwischen diesen beiden Funktionen beliebig klein gemacht werden kann. <\/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 es existieren zu jedem <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> stetige Funktionen <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <\/math> wie in der Proposition. Sei <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und w\u00e4hle stetige Funktionen <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/math> und <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <\/math> mit <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/math> und                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msup><\/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\">Da <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <\/math> als stetige Funktion nach Satz&nbsp;<a href=\"..\/..\/chapter\/integrierbarkeit-stetiger-funktionen#x1-123001r42\">4.42<\/a> auch Riemann-integrierbar ist, so existiert eine Treppenfunktion <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\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>o<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>o<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/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> Genauso existiert <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> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <\/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> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><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> Zusammenfassend gilt <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>o<\/mi><\/math> sowie nach Linearit\u00e4t <\/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\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>o<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow> <mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><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> <mn>3<\/mn><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\">Da <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig war, beweist dies mit Hilfe von Proposition <a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110005r12\">4.12<\/a> die erste, einfachere Richtung der Proposition. <\/p><p class=\"indent\">Wir beweisen nun die verbleibende Richtung und nehmen dazu zuerst an, dass <math display=\"inline\"><mi>t<\/mi><\/math> eine Treppenfunktion ist. Sei <math display=\"inline\"><mi>\u2128<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> eine Zerlegung in Konstanzintervalle f\u00fcr <math display=\"inline\"><mi>t<\/mi><\/math> und seien <math display=\"inline\"><msub><mrow><mi>c<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> die Konstanzwerte. Wir konstruieren eine stetige Funktion <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> mit <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <\/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> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wenden wir dies auf <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><\/math> an, so l\u00e4sst sich analog                                                                                                                                                                           eine stetige Funktion <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/math> auf <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><\/math> und <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/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> finden. <\/p><p class=\"indent\">Bevor wir eine explizite Definition von <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/math> angeben, fassen wir die Idee in folgendem Bild zusammen. <\/p> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=406&amp;url=Pictures\/R-integral\/sandwich\/trpfkt-sandwich.pdf\"><img id=\"z669b81786b73\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/R-integral\/sandwich\/trpfkt-sandwich.svg\" width=\"406\"><\/div> <a id=\"x1-124002r4\"><\/a> <a id=\"x1-124003\"><\/a> <br><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;4.4:       <\/span><span class=\"content\">In       blau       gestrichelt       die       Treppenfunktion               <span class=\"maperiod\"><math display=\"inline\"><mi>t<\/mi><\/math><\/span><span class=\"period\">,<\/span>               in      rot      eine      stetige,      st\u00fcckweise      lineare      Funktion               <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span>               F\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span>               die       nicht       in       der       N\u00e4he       eines       Punktes       von               <math display=\"inline\"><mi>\u2128<\/mi><\/math>               sind, ist <span class=\"maperiod\"><math display=\"inline\"><mi>t<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span>               In der N\u00e4he der Punkte von <math display=\"inline\"><mi>\u2128<\/mi><\/math>               ersetzen wir <math display=\"inline\"><mi>t<\/mi><\/math>             durch   eine   gen\u00fcgend   hohe   aber   \u201estetige   Spitze\u201c,   so   dass               <math display=\"inline\"><mi>t<\/mi><\/math>             unter der Spitze liegt. Damit die Definition etwas einfacher ist, w\u00e4hlen               wir \u00fcberall die gleiche \u201eH\u00f6he und Breite\u201c  der Spitze; nat\u00fcrlich ist               dies nicht zwingend n\u00f6tig.                                                          &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <p class=\"indent\">Sei <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> mit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\">min<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">{<\/mo><mo class=\"MathClass-rel\">|<\/mo><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><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">\u2026<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-close\">}<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">vorerst beliebig (dies wird die \u201ehalbe Breite\u201c  der Spitzen sein). Da <math display=\"inline\"><mi>t<\/mi><\/math> eine Treppenfunktion ist, existeren <math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> min<\/mi><mo>  <\/mo><mi>t<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mi>t<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Zur formalen Definition von <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <\/math> verwenden wir die Zerlegung <span class=\"maperiod\"><math display=\"inline\"><mi>\u2128<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">,<\/span> die Konstanzwerte <math display=\"inline\"><msub><mrow><mi>c<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> von <math display=\"inline\"><mi>t<\/mi><\/math> und f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> die Fallunterscheidung <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow class=\"cases\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mi>M<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>M<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mfrac><mrow> <mi>x<\/mi><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> <mrow><mi>\u03b4<\/mi><\/mrow><\/mfrac> <mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><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><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mi>M<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>M<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mfrac><mrow> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow> <mrow><mi>\u03b4<\/mi><\/mrow><\/mfrac> <mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mi>M<\/mi> <mspace class=\"quad\" width=\"1em\" \/> <\/mtd> <mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle> <mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wobei <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> beliebig ist und die verschiedenen F\u00e4lle auf Grund unserer Wahl von <math display=\"inline\"><mi>\u03b4<\/mi><\/math> disjunkt sind. Aus Korollar&nbsp;<a href=\"..\/..\/chapter\/stetigkeit#x1-94010r51\">3.51<\/a> und mehrfacher Anwendung von \u00dcbung&nbsp;<a href=\"..\/..\/chapter\/stetigkeit#x1-94013r54\">3.54<\/a> folgt, dass <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <\/math> stetig ist. Dank der Definition von <math display=\"inline\"><mi>m<\/mi><\/math> und <math display=\"inline\"><mi>M<\/mi><\/math> ist auch <span class=\"maperiod\"><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>t<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>M<\/mi><\/math><\/span><span class=\"period\">.<\/span> Weiter gilt f\u00fcr jedes Konstanzintervall <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> dass <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>t<\/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> f\u00fcr alle <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> mit Distanz gr\u00f6sser als <math display=\"inline\"><mi>\u03b4<\/mi><\/math> von den Randpunkten <span class=\"maperiod\"><math display=\"inline\"><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><\/math><\/span><span class=\"period\">.<\/span> Insbesondere ist                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/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><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-bin\">+<\/mo><mi>\u03b4<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow> <mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u03b4<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>M<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>m<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>\u03b4<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>M<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>m<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><mi>M<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>m<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>\u03b4<\/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\">und damit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow> <mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>2<\/mn> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>M<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>m<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>\u03b4<\/mi><mi>n<\/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\">W\u00e4hlen wir nun <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><mi>M<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>m<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>n<\/mi><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> so ist <span class=\"maperiod\"><math display=\"inline\"><mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><mi>M<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>m<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>\u03b4<\/mi><mi>n<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span> Daher hat die stetige Funktion <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/math> alle gew\u00fcnschten Eigenschaften. <\/p><p class=\"indent\">Sei nun <math display=\"inline\"><mi>f<\/mi><\/math> eine beliebige Riemann-integrierbare Funkton und sei <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> Nach Proposition&nbsp;<a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110005r12\">4.12<\/a> existieren Treppenfunktionen <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>o<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> mit <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>o<\/mi><\/math> 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> <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> Seien <math display=\"inline\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>o<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>C<\/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> wie oben konstruiert mit <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>u<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>o<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>o<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>u<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/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> 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> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>o<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>o<\/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> Die stetigen Funktionen <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>u<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/math>                                                                                                                                                                           und <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>o<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/math> erf\u00fcllen damit <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/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> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/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> <mn>3<\/mn><mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span> Da <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig war, haben wir damit die zweite Implikation der Proposition gezeigt. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <a id=\"x1-124004r123\"><\/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: 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{text-align:center;}\ndiv.thanks{text-align:left; margin-left:10%; font-size:85%; font-style:italic; }\ndiv.author{white-space: nowrap;}\n.quotation {margin-bottom:0.25em; margin-top:0.25em; margin-left:1em; }\n.abstract p {margin-left:5%; margin-right:5%;}\ndiv.abstract {width:100%;}\ndiv.tabular, div.center div.tabular {text-align: center; margin-top:0.5em; margin-bottom:0.5em; }\ntable.tabular td p{margin-top:0em;}\ntable.tabular {margin-left: auto; margin-right: auto;}\ntd p:first-child{ margin-top:0em; }\ntd p:last-child{ margin-bottom:0em; }\ndiv.td00{ margin-left:0pt; margin-right:0pt; }\ndiv.td01{ margin-left:0pt; margin-right:5pt; }\ndiv.td10{ margin-left:5pt; margin-right:0pt; }\ndiv.td11{ margin-left:5pt; margin-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\ntd.td00{ padding-left:0pt; padding-right:0pt; }\ntd.td01{ padding-left:0pt; padding-right:5pt; }\ntd.td10{ padding-left:5pt; padding-right:0pt; }\ntd.td11{ padding-left:5pt; padding-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\n.hline hr, .cline hr{ height : 0px; margin:0px; }\n.hline td, .cline td{ padding: 0; }\n.hline hr, .cline hr{border:none;border-top:1px solid black;}\n.equation-star td{text-align:center; vertical-align:middle; }\ntable.equation-star { width:100%; border-bottom-color: rgb(255,255,255); }\n#content table.equation-star, #content table.equation-star tbody tr td { border: 0px none rgb(255,255,255); }\nmtd.align-odd{margin-left:2em; text-align:right;}\nmtd.align-even{margin-right:2em; text-align:left;}\n.boxed{border: 1px solid black; padding-left:2px; padding-right:2px;}\n.rotatebox{display: inline-block;}\n.item-head{float:left;width:2em;clear:left;}\n.item-content{margin-left:2em;}\n .foreignobject {line-height:100%; font-size:120%; font-family:STIXgeneral,Times,Symbol,cmr10,CMSY10,CMEX10;padding:0; margin:0; text-align:center; }\nmath {vertical-align:baseline; line-height:100%; font-size:100%; font-family:STIXGeneral,Times,Symbol, cmr10,cmsy10,cmex10,cmmi10; font-style: normal; margin:0; padding:0; }\n\n.entry-title{display: none}\n\ndiv.newtheorem { margin-bottom: 2em; margin-top: 2em; border: 1px solid #333; background: #c7e4da; border-color: #4eb79e;}\ndiv.newtheorem h3 { background: #4eb79e; color: white; padding: 0px 15px 0px 15px; margin-top: 12px}\ndiv.newtheorem p { padding: 15px 15px 15px 15px; }\n\ndiv.newtheorem p span.head .ecbx-1095{font-weight: bold}\ndiv.newtheorem p .ecti-1095{font-style: italic}\ndiv.newtheorem div.custom-itemize{font-style: italic}\ndiv.quote{font-style: italic}\ndiv.newtheorem dl, dl.enumerate {display: grid; grid-template-columns: 5% auto; align-items: start; margin-top: 1em}\ndiv.newtheorem dl dd, dl.enumerate dd {margin-bottom: 0.5em}\ndiv.newtheorem dl dt, dl.enumerate dt {font-weight: normal; margin-top: 0px; text-align: right; margin-right: 15%}\ndiv.newtheorem dl dd {font-style: italic}\ndiv.newtheorem dl dt {font-style: italic}\ndiv.proof p span.ecti-1095 {font-style: italic}\ndiv.figure p img { margin-left: auto; margin-right: auto; display: block; }\ndiv.mefigcentered, div.figure { text-align: center }\n\ndl:after {content:\"\";display:table;clear:both;}\ndd {padding:.5em 0;}\ndl {width:100%;}\ndt, dd {display:inline-block; 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 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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=\"zd64c49abb2e0\" class=\"sectionHead\"><span class=\"titlemark\">4.7 <\/span> <a id=\"x1-1230007\"><\/a>Integrierbarkeit stetiger Funktionen<\/h3> <p class=\"noindent\">Aus Abschnitt <a href=\"..\/..\/chapter\/integration-von-polynomen#x1-1220006\">4.6<\/a> wissen wir bereits, dass Polynomfunktionen integrierbar sind. In diesem Abschnitt m\u00f6chten wir nun unter Verwendung der Beschr\u00e4nktheit und der gleichm\u00e4ssigen Stetigkeit stetiger Funktionen auf kompakten Intervallen (Satz <a href=\"..\/..\/chapter\/stetige-funktionen-auf-kompakten-intervallen#x1-100001r69\">3.69<\/a> respektive Satz <a href=\"..\/..\/chapter\/stetige-funktionen-auf-kompakten-intervallen#x1-102005r77\">3.77<\/a>) folgendes allgemeines Resultat beweisen. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z4c4e159b10fa\"> <a id=\"x1-123001r42\"><\/a> <span class=\"ecbx-1095\">Satz 4.42 <\/span>(Stetige Funktionen und das Riemann-Integral)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Eine stetige Funktion auf einem kompakten Intervall <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">mit<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">ist Riemann-integrierbar.<\/span> <\/p> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Sei <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>C<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Nach Satz <a href=\"..\/..\/chapter\/stetige-funktionen-auf-kompakten-intervallen#x1-102005r77\">3.77<\/a> ist <math display=\"inline\"><mi>f<\/mi><\/math> gleichm\u00e4ssig stetig und es gibt ein <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> so dass f\u00fcr alle <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> gilt <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <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\"><mstyle class=\"label\" id=\"x1-123002r16\" \/><mstyle class=\"maketag\"><mtext>(4.16)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Sei <math display=\"inline\"><mi>\u2128<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> eine Zerlegung von <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">\u2026<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><\/munder> <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\">&lt;<\/mo> <mi>\u03b4<\/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\">Zum Beispiel k\u00f6nnen wir die Zerlegung durch <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>k<\/mi> <mfrac> <mrow> <mi>b<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>a<\/mi><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/math> f\u00fcr <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi><\/math> und ein hinreichend grosses <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> definieren. Wir definieren f\u00fcr jedes <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> die Zahlen <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> min<\/mi><mo>  <\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/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><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/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><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wobei wir Korollar <a href=\"..\/..\/chapter\/stetige-funktionen-auf-kompakten-intervallen#x1-101001r71\">3.71<\/a> f\u00fcr die Existenz von Minimum und Maximum verwendet haben. Wir behaupten nun, dass f\u00fcr alle <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> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <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\">gilt. In der Tat ist <math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi class=\"qopname\">min<\/mi><mo>  <\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi class=\"qopname\">min<\/mi><mo>  <\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo> <\/mrow> <\/msub> <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><span class=\"period\">.<\/span> Da aber <math display=\"inline\"><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> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><\/math> ist, haben wir auch <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi class=\"qopname\">min<\/mi><mo>  <\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><\/math><\/span><span class=\"period\">.<\/span> Auf Grund der Wahl von <math display=\"inline\"><mi>\u03b4<\/mi><\/math> mit (<a href=\"..\/..\/chapter\/integrierbarkeit-stetiger-funktionen#x1-123002r16\">4.16<\/a>) erhalten wir also unsere Behauptung <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi class=\"qopname\">min<\/mi><mo>  <\/mo><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Wir definieren nun Treppenfunktionen <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>o<\/mi><\/math> durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>u<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><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><mstyle class=\"text\"><mtext>&nbsp;f\u00fcr&nbsp;<\/mtext><\/mstyle><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><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi> <\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/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\"><mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><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><mstyle class=\"text\"><mtext>&nbsp;f\u00fcr&nbsp;<\/mtext><\/mstyle><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><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi> <\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> Nach Definition von <math display=\"inline\"><msub><mrow><mi>m<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>M<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/math> f\u00fcr <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> gilt daher <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> Des Weiteren ist                                                                                                                                                                           <\/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\">=<\/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>M<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>m<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <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\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><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> <mi>\ud835\udf00<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>b<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Da <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig war (und <math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/math> fix ist), zeigt dies mittels Proposition <a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110005r12\">4.12<\/a>, dass <math display=\"inline\"><mi>f<\/mi><\/math> Riemann-integrierbar ist. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z8a6bf1f311ec\"> <a id=\"x1-123003r43\"><\/a> <span class=\"ecbx-1095\">Applet 4.43 <\/span>(Integrierbarkeit einer \u201ezittrigen\u201c Funktion)<span class=\"ecbx-1095\">.<\/span> <\/h4> <div class=\"wp-nocaption \"><\/div><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"652px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/ms4mjkun\/width\/688\/height\/652\/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, dass eine stetige aber zittrige Funktion wie im dargestellten Graphen auch<\/span> <span class=\"ecti-1095\">Riemann-integrierbar ist.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Anmerkung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Was wir auch mitunter sehen k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen, ist, dass geogebra mit der verwendeten<\/span> <span class=\"ecti-1095\">Funktion manchmal  Problem  hat  und  manche  der  dargestellten  Untersummen  oder<\/span> <span class=\"ecti-1095\">Obersummen eigentlich nicht richtig dargestellt und berechnet werden. Unabh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngig davon<\/span> <span class=\"ecti-1095\">haben wir aber in unserem Beweis schon die Riemann-Integrierbarkeit gesehen, sind also f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">die gew<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nschte Aussage nicht auf geogebra angewiesen.<\/span><\/p><\/details>  <\/div> <a id=\"x1-123004r120\"><\/a> <h4 id=\"z5e8a27086cff\" class=\"subsectionHead\"><span class=\"titlemark\">4.7.1 <\/span> <a id=\"x1-1240001\"><\/a>Sandwich mittels stetigen Funktionen<\/h4> <p class=\"noindent\">Die folgende Proposition zeigt, dass auch allgemeine Riemann-integrierbare Funktionen einen gewissen Zusammenhang zu dem Begriff der Stetigkeit aufweisen. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z778781f18003\"> <a id=\"x1-124001r44\"><\/a> <span class=\"ecbx-1095\">Proposition 4.44 <\/span>(Sandwich-Kriterium mit stetigen Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Eine Funktion <\/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\">auf<\/span> <span class=\"ecti-1095\">einem kompakten Intervall <\/span><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">ist genau dann<\/span> <span class=\"ecti-1095\">Riemann-integrierbar, wenn 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\">stetige Funktionen <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">existieren mit <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/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> <\/div> <p class=\"indent\">Wir empfehlen Ihnen sich dies geometrisch vorzustellen: <math display=\"inline\"><mi>f<\/mi><\/math> wird von den stetigen Funktionen <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/math> und <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <\/math> eingez\u00e4unt, wobei die Fl\u00e4che zwischen diesen beiden Funktionen beliebig klein gemacht werden kann. <\/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 es existieren zu jedem <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> stetige Funktionen <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <\/math> wie in der Proposition. Sei <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und w\u00e4hle stetige Funktionen <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/math> und <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <\/math> mit <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/math> und                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msup><\/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\">Da <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <\/math> als stetige Funktion nach Satz&nbsp;<a href=\"..\/..\/chapter\/integrierbarkeit-stetiger-funktionen#x1-123001r42\">4.42<\/a> auch Riemann-integrierbar ist, so existiert eine Treppenfunktion <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\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>o<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>o<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/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> Genauso existiert <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> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <\/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> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><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> Zusammenfassend gilt <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>o<\/mi><\/math> sowie nach Linearit\u00e4t <\/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\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>o<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow> <mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><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> <mn>3<\/mn><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\">Da <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig war, beweist dies mit Hilfe von Proposition <a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110005r12\">4.12<\/a> die erste, einfachere Richtung der Proposition. <\/p><p class=\"indent\">Wir beweisen nun die verbleibende Richtung und nehmen dazu zuerst an, dass <math display=\"inline\"><mi>t<\/mi><\/math> eine Treppenfunktion ist. Sei <math display=\"inline\"><mi>\u2128<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> eine Zerlegung in Konstanzintervalle f\u00fcr <math display=\"inline\"><mi>t<\/mi><\/math> und seien <math display=\"inline\"><msub><mrow><mi>c<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> die Konstanzwerte. Wir konstruieren eine stetige Funktion <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> mit <math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <\/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> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wenden wir dies auf <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi><\/math> an, so l\u00e4sst sich analog                                                                                                                                                                           eine stetige Funktion <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/math> auf <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><\/math> und <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/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> finden. <\/p><p class=\"indent\">Bevor wir eine explizite Definition von <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/math> angeben, fassen wir die Idee in folgendem Bild zusammen. <\/p> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=406&amp;url=Pictures\/R-integral\/sandwich\/trpfkt-sandwich.pdf\"><img decoding=\"async\" id=\"z669b81786b73\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/R-integral\/sandwich\/trpfkt-sandwich.svg\" width=\"406\" \/><\/div> <a id=\"x1-124002r4\"><\/a> <a id=\"x1-124003\"><\/a> <br \/><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;4.4:       <\/span><span class=\"content\">In       blau       gestrichelt       die       Treppenfunktion               <span class=\"maperiod\"><math display=\"inline\"><mi>t<\/mi><\/math><\/span><span class=\"period\">,<\/span>               in      rot      eine      stetige,      st\u00fcckweise      lineare      Funktion               <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span>               F\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span>               die       nicht       in       der       N\u00e4he       eines       Punktes       von               <math display=\"inline\"><mi>\u2128<\/mi><\/math>               sind, ist <span class=\"maperiod\"><math display=\"inline\"><mi>t<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span>               In der N\u00e4he der Punkte von <math display=\"inline\"><mi>\u2128<\/mi><\/math>               ersetzen wir <math display=\"inline\"><mi>t<\/mi><\/math>             durch   eine   gen\u00fcgend   hohe   aber   \u201estetige   Spitze\u201c,   so   dass               <math display=\"inline\"><mi>t<\/mi><\/math>             unter der Spitze liegt. Damit die Definition etwas einfacher ist, w\u00e4hlen               wir \u00fcberall die gleiche \u201eH\u00f6he und Breite\u201c  der Spitze; nat\u00fcrlich ist               dies nicht zwingend n\u00f6tig.                                                          &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <p class=\"indent\">Sei <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> mit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\">min<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">{<\/mo><mo class=\"MathClass-rel\">|<\/mo><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><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">\u2026<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">}<\/mo><mo class=\"MathClass-close\">}<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">vorerst beliebig (dies wird die \u201ehalbe Breite\u201c  der Spitzen sein). Da <math display=\"inline\"><mi>t<\/mi><\/math> eine Treppenfunktion ist, existeren <math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> min<\/mi><mo>  <\/mo><mi>t<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> max<\/mi><mo>  <\/mo> <mi>t<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Zur formalen Definition von <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <\/math> verwenden wir die Zerlegung <span class=\"maperiod\"><math display=\"inline\"><mi>\u2128<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">,<\/span> die Konstanzwerte <math display=\"inline\"><msub><mrow><mi>c<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> von <math display=\"inline\"><mi>t<\/mi><\/math> und f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> die Fallunterscheidung <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow class=\"cases\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mi>M<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>M<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mfrac><mrow> <mi>x<\/mi><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> <mrow><mi>\u03b4<\/mi><\/mrow><\/mfrac> <mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><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><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mi>M<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>M<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mfrac><mrow> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><\/mrow> <mrow><mi>\u03b4<\/mi><\/mrow><\/mfrac> <mspace class=\"quad\" width=\"1em\" \/><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mi>M<\/mi> <mspace class=\"quad\" width=\"1em\" \/> <\/mtd> <mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle> <mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wobei <math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">{<\/mo><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> beliebig ist und die verschiedenen F\u00e4lle auf Grund unserer Wahl von <math display=\"inline\"><mi>\u03b4<\/mi><\/math> disjunkt sind. Aus Korollar&nbsp;<a href=\"..\/..\/chapter\/stetigkeit#x1-94010r51\">3.51<\/a> und mehrfacher Anwendung von \u00dcbung&nbsp;<a href=\"..\/..\/chapter\/stetigkeit#x1-94013r54\">3.54<\/a> folgt, dass <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <\/math> stetig ist. Dank der Definition von <math display=\"inline\"><mi>m<\/mi><\/math> und <math display=\"inline\"><mi>M<\/mi><\/math> ist auch <span class=\"maperiod\"><math display=\"inline\"><mi>m<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>t<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>M<\/mi><\/math><\/span><span class=\"period\">.<\/span> Weiter gilt f\u00fcr jedes Konstanzintervall <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> dass <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>t<\/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> f\u00fcr alle <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> mit Distanz gr\u00f6sser als <math display=\"inline\"><mi>\u03b4<\/mi><\/math> von den Randpunkten <span class=\"maperiod\"><math display=\"inline\"><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><\/math><\/span><span class=\"period\">.<\/span> Insbesondere ist                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/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><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-bin\">+<\/mo><mi>\u03b4<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow> <mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-bin\">\u2212<\/mo><mi>\u03b4<\/mi><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>c<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>M<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>m<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>\u03b4<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>M<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>m<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><mi>M<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>m<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>\u03b4<\/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\">und damit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow> <mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo>\u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><\/mrow><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>t<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>2<\/mn> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>M<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>m<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>\u03b4<\/mi><mi>n<\/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\">W\u00e4hlen wir nun <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><mi>M<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>m<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>n<\/mi><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> so ist <span class=\"maperiod\"><math display=\"inline\"><mn>2<\/mn><mo class=\"MathClass-open\">(<\/mo><mi>M<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>m<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>\u03b4<\/mi><mi>n<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span> Daher hat die stetige Funktion <math display=\"inline\"><msub><mrow><mi>t<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/math> alle gew\u00fcnschten Eigenschaften. <\/p><p class=\"indent\">Sei nun <math display=\"inline\"><mi>f<\/mi><\/math> eine beliebige Riemann-integrierbare Funkton und sei <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> Nach Proposition&nbsp;<a href=\"..\/..\/chapter\/definition-des-riemann-integrals#x1-110005r12\">4.12<\/a> existieren Treppenfunktionen <math display=\"inline\"><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>o<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\ud835\udcaf<\/mi><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> mit <math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>o<\/mi><\/math> 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> <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> Seien <math display=\"inline\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>o<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>C<\/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> wie oben konstruiert mit <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>u<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>u<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>o<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>o<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>u<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>u<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/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> 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> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>o<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>o<\/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> Die stetigen Funktionen <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>u<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/math>                                                                                                                                                                           und <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>o<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/math> erf\u00fcllen damit <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub><\/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> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/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> <mn>3<\/mn><mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span> Da <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig war, haben wir damit die zweite Implikation der Proposition gezeigt. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <a id=\"x1-124004r123\"><\/a> 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