{"id":72,"date":"2021-12-15T09:53:15","date_gmt":"2021-12-15T09:53:15","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/riemann-summen\/"},"modified":"2021-12-15T09:53:15","modified_gmt":"2021-12-15T09:53:15","slug":"riemann-summen","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/riemann-summen\/","title":{"raw":"Riemann-Summen","rendered":"Riemann-Summen"},"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=\"zde44a1c6de94\" class=\"sectionHead\"><span class=\"titlemark\">6.5 <\/span> <a id=\"x1-1800005\"><\/a>Riemann-Summen<\/h3> <p class=\"noindent\">Riemann gab <math display=\"inline\"><mn>1<\/mn><mn>8<\/mn><mn>5<\/mn><mn>4<\/mn><\/math> eine formale Definition des Integrals mit Hilfe sogenannter Riemann-Summen und eines \u201eGrenz\u00fcbergangs\u201c, dessen Definition \u00c4hnlichkeiten zu den Definitionen des Grenzwertes einer Folge und des Grenzwertes einer Funktion aufweist. Wie in Kapitel <a href=\"..\/..\/part\/das-riemann-integral#x1-1060004\">4<\/a> sei im Folgenden <math display=\"inline\"><mi>f<\/mi><\/math> eine reellwertige Funktion auf einem kompakten Intervall <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z6b4e9e8cd269\"> <a id=\"x1-180001r46\"><\/a> <span class=\"ecbx-1095\">Definition 6.46 <\/span>(Riemann-Summen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">F\u00fcr eine Zerlegung <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> 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> definieren wir die <span class=\"ecbx-1095\">Maschenweite der Zerlegung <\/span><math display=\"inline\"><mi>\u2128<\/mi><\/math> als <span class=\"maperiod\"><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>\u2128<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><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><\/math><\/span><span class=\"period\">.<\/span> Weiters bezeichnen wir <math display=\"inline\"><mstyle><mi>z<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> als eine <span class=\"ecbx-1095\">erlaubte Wahl von Zwischenpunkten <\/span>der Zerlegung&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi>\u2128<\/mi><\/math><\/span><span class=\"period\">,<\/span> falls <math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi> <\/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> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> F\u00fcr eine reellwertige Funktion&nbsp;<math display=\"inline\"><mi>f<\/mi><\/math> auf <span class=\"maperiod\"><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><span class=\"period\">,<\/span> eine Zerlegung&nbsp;<math display=\"inline\"><mi>\u2128<\/mi><\/math> von&nbsp;<math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> und eine erlaubte Wahl von Zwischenpunkten&nbsp;<math display=\"inline\"><mstyle><mi>z<\/mi><\/mstyle><\/math> definieren wir die <span class=\"ecbx-1095\">Riemann-Summe <\/span>durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>R<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>z<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo> <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><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/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-close\">)<\/mo><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\">Das heisst, wir betrachten beliebige Punkte <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi> <\/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> die Funktionswerte <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> an diesen Punkten und hoffen, dass diese halbwegs repr\u00e4sentativ f\u00fcr die Funktionswerte von <math display=\"inline\"><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><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> <\/mrow> <\/msub> <\/math> sind. Diese Hoffnung mag zwar nicht in allen Teilintervallen immer zutreffen, trotzdem ist die Riemann-Summe eine Approximation des Riemann-Integrals in folgendem Sinne. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z207e86d09555\"> <a id=\"x1-180002r47\"><\/a> <span class=\"ecbx-1095\">Satz 6.47 <\/span>(Riemann-Integral \u00fcber Riemann-Summen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">eine Riemann-integrierbare<\/span> <span class=\"ecti-1095\">reellwertige Funktion auf <\/span><span class=\"maperiod\"><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><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann ist <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">der Grenzwert<\/span> <span class=\"ecti-1095\">der Riemann-Summen <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>R<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">wenn die Maschenweite <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">der Zerlegung gegen Null geht. Formal geschrieben gilt also<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\u2128<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mstyle><mi>z<\/mi><\/mstyle> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>\u2128<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <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\" \/> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>R<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>z<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>\u2128<\/mi><\/math> <span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber die<\/span> <span class=\"ecti-1095\">Zerlegungen von <\/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\">l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">uft und<\/span> <math display=\"inline\"><mstyle><mi>z<\/mi><\/mstyle><\/math> <span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber die erlaubten Wahlen von<\/span> <span class=\"ecti-1095\">Zwischenpunkten der Zerlegung<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u2128<\/mi><\/math> <span class=\"ecti-1095\">(wie in Definition <\/span><a href=\"..\/..\/chapter\/riemann-summen#x1-180001r46\"><span class=\"ecti-1095\">6.46<\/span><\/a><span class=\"ecti-1095\">) l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">uft.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z8f993e43a7a7\"> <span class=\"ecti-1095\">Bemerkung.<\/span><\/h4> <p class=\"indent\">Die Konvergenz der Riemann-Summen wie in obigem Satz ist sogar eine Charakterisierung der Riemann-Integrierbarkeit \u2013 siehe \u00dcbung&nbsp;<a href=\"..\/..\/chapter\/riemann-summen#x1-180004r49\">6.49<\/a>. Es gibt auch noch weitere, \u00e4quivalente Bedingungen, aber wir begn\u00fcgen uns mit der Aussage in Satz <a href=\"..\/..\/chapter\/riemann-summen#x1-180002r47\">6.47<\/a>. <\/p> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis von Satz <a href=\"..\/..\/chapter\/riemann-summen#x1-180002r47\">6.47<\/a>.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\"> Wir beweisen den Satz zuerst unter der zus\u00e4tzlichen Annahme, dass <math display=\"inline\"><mi>f<\/mi><\/math> stetig ist. Dank dem Sandwich-Kriterium f\u00fcr Riemann-Integrierbarkeit aus Proposition&nbsp;<a href=\"..\/..\/chapter\/integrierbarkeit-stetiger-funktionen#x1-124001r44\">4.44<\/a> wird sich dies als ausreichend herausstellen. <\/p><p class=\"indent\">Sei also <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> stetig 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&nbsp;<a href=\"..\/..\/chapter\/stetige-funktionen-auf-kompakten-intervallen#x1-102005r77\">3.77<\/a> ist <math display=\"inline\"><mi>f<\/mi><\/math> gleichm\u00e4ssig stetig, womit <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> existiert mit <math display=\"inline\"><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><\/math> 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> welche <math display=\"inline\"><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><\/math> erf\u00fcllen. 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 mit Maschenweite <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><\/math> und <math display=\"inline\"><mstyle><mi>z<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> eine erlaubte Wahl von Zwischenpunkten. Dann gilt f\u00fcr alle <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> wegen der Dreiecksungleichung f\u00fcr das Riemann-Integral in Satz&nbsp;<a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-113002r24\">4.24<\/a> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><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><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>z<\/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><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><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><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>z<\/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><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo><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><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/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\">&lt;<\/mo> <mi>\ud835\udf00<\/mi> <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-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\">In der Tat is <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/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> <math display=\"inline\"><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-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><\/math> wegen <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><\/math> und somit <math display=\"inline\"><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><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Insbesondere ist <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>R<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>z<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"> <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>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><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><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>z<\/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><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">&lt;<\/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><mi>\ud835\udf00<\/mi> <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><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Da <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> die Zerlegung <math display=\"inline\"><mi>\u2128<\/mi><\/math> mit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><\/math> sowie die erlaubten Zwischenpunkte <math display=\"inline\"><mstyle><mi>z<\/mi><\/mstyle><\/math> beliebig waren, beweist dies die Proposition f\u00fcr alle stetigen Funktionen <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Sei nun <math display=\"inline\"><mi>f<\/mi><\/math> eine beliebige Riemann-integrierbare Funktion 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\/integrierbarkeit-stetiger-funktionen#x1-124001r44\">4.44<\/a> existieren <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/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> 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 <span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><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><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> Nach der oben schon bewiesenen Aussage sei <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> mit der Eigenschaft, dass f\u00fcr alle Zerlegungen <math display=\"inline\"><mi>\u2128<\/mi><\/math> mit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><\/math> und f\u00fcr alle erlaubten Zwischenpunkte <math display=\"inline\"><mstyle><mi>z<\/mi><\/mstyle><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>R<\/mi> <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-punc\">,<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>z<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <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><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und genauso f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Nun gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>R<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>z<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>R<\/mi> <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-punc\">,<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>z<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <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><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">&lt;<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>\ud835\udf00<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und unter Verwendung von <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/math> auf \u00e4hnliche Weise <span class=\"maperiod\"><math display=\"inline\"><mi>R<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>z<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">&gt;<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span> Damit ist der Satz bewiesen. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z63da9ff1d63a\"> <a id=\"x1-180003r48\"><\/a> <span class=\"ecbx-1095\">Applet 6.48 <\/span>(Riemann-Summen f\u00fcr die Parabel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"532px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/gdcuwv7x\/width\/688\/height\/532\/border\/888888\/rc\/false\/ai\/false\/sdz\/true\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir sehen Riemann-Summen f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Parabel aus Abschnitt <\/span><a href=\"..\/..\/chapter\/quadratur-der-parabel#x1-40001\"><span class=\"ecti-1095\">1.1<\/span><\/a><span class=\"ecti-1095\">, wobei die Zwischenpunkte<\/span> <span class=\"ecti-1095\">zuf<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">llig gew<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlt werden.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z5de9c3e879fc\"> <a id=\"x1-180004r49\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 6.49 <\/span>(Charakterisierung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">reellen Zahlen und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie die Umkehrung zu Satz <\/span><a href=\"..\/..\/chapter\/riemann-summen#x1-180002r47\"><span class=\"ecti-1095\">6.47<\/span><\/a><span class=\"ecti-1095\">, also dass die Konvergenz der Riemann-Summen f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">zu einer Zahl <\/span><math display=\"inline\"><mi>I<\/mi><\/math> <span class=\"ecti-1095\">(wie in Satz <\/span><a href=\"..\/..\/chapter\/riemann-summen#x1-180002r47\"><span class=\"ecti-1095\">6.47<\/span><\/a><span class=\"ecti-1095\">) die Riemann-Integrierbarkeit von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">und die Gleichung <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>I<\/mi><\/math> <span class=\"ecti-1095\">impliziert.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">W<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlen Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ein <\/span><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">wie in der Konvergenz der Riemann-Summen. Sei <\/span><math display=\"inline\"><mi>\u2128<\/mi><\/math> <span class=\"ecti-1095\">eine Zerlegung mit Maschenweite <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Betrachten Sie nun das Infimum und das Supremum aller Riemannsummen zu erlaubten<\/span> <span class=\"ecti-1095\">Wahlen von Zwischenpunkten der Zerlegung <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2128<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">und verkn<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">pfen Sie dies mit einer Untersumme und einer Obersumme.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z431171e7ef4d\"> <a id=\"x1-180005r50\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 6.50 <\/span>(Riemann-Integrierbarkeit stetiger Funktion)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Nach <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/riemann-summen#x1-180004r49\"><span class=\"ecti-1095\">6.49<\/span><\/a> <span class=\"ecti-1095\">h<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tten wir die Konvergenz der Riemann-Summen als Definition f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">Riemann-integrierbare Funktionen verwenden k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen. Zeigen Sie, dass stetige Funktionen<\/span> <span class=\"ecti-1095\">Riemann-integrierbar sind in diesem Sinne (ohne Satz<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/integrierbarkeit-stetiger-funktionen#x1-123001r42\"><span class=\"ecti-1095\">4.42<\/span><\/a> <span class=\"ecti-1095\">zu verwenden).<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Hinweis: Betrachten Sie hierzu zuerst eine Folge von Zerlegungen <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u2128<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">mit der Eigenschaft, dass <\/span><math display=\"inline\"><msub><mrow><mi>\u2128<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">feiner ist als <\/span><math display=\"inline\"><msub><mrow><mi>\u2128<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Da uns der Grenzwert einer Folge von Riemann-Summen a priori nicht bekannt ist, kann man<\/span> <span class=\"ecti-1095\">eher zeigen, dass die Folge eine Cauchy-Folge ist.<\/span> <\/p> <\/div> <a id=\"x1-180006r179\"><\/a> <h4 id=\"z64d40a3d21e2\" class=\"subsectionHead\"><span class=\"titlemark\">6.5.1 <\/span> <a id=\"x1-1810001\"><\/a>Vektorwertige Integrale<\/h4> <p class=\"noindent\">Unsere urspr\u00fcngliche Definition des Riemann-Integrals in Kapitel <a href=\"..\/..\/part\/das-riemann-integral#x1-1060004\">4<\/a> verwendete die Ungleichung <math display=\"inline\"><mo class=\"MathClass-rel\">\u2264<\/mo><\/math> in                                                                                                                                                                           <math display=\"inline\"><mi>\u211d<\/mi><\/math> in zentraler Weise und kann deswegen nicht auf diese Weise f\u00fcr vektorwertige Funktionen verallgemeinert werden. Riemann-Summen lassen sich hingegen leicht verallgemeinern. F\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>f<\/mi><\/mstyle> <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> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <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> und <math display=\"inline\"><mstyle><mi>z<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> eine zul\u00e4ssige Wahl von Zwischenpunkten gem\u00e4ss Definition <a href=\"..\/..\/chapter\/riemann-summen#x1-180001r46\">6.46<\/a> setzen wir wie zuvor <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>R<\/mi><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>f<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>z<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo> <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><mstyle><mi>f<\/mi><\/mstyle><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/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-close\">)<\/mo><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\">Des Weiteren sagen wir, dass <math display=\"inline\"><mstyle><mi>f<\/mi><\/mstyle> <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> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> Riemann-integrierbar ist, falls <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mi>f<\/mi><\/mstyle> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <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> und die Komponentenfunktionen <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>j<\/mi><\/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> f\u00fcr <math display=\"inline\"><mi>j<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><\/math> Riemann-integrierbar sind (man vergleiche dies zu Proposition&nbsp;<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-148002r44\">5.44<\/a>). Das Riemann-Integral wird dann komponentenweise definiert durch                                                                                                                                                                           <\/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><mstyle><mi>f<\/mi><\/mstyle> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mo>\u2026<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mo class=\"MathClass-punc\">,<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mi>d<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup><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\">Satz <a href=\"..\/..\/chapter\/riemann-summen#x1-180002r47\">6.47<\/a> gilt nun analog f\u00fcr Riemann-integrierbare Funktionen 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> nach <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math><\/span><span class=\"period\">.<\/span> In der Tat gilt <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mstyle><mi>f<\/mi><\/mstyle> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>R<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mstyle><mi>f<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> genau dann, wenn f\u00fcr alle <math display=\"inline\"><mi>j<\/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>d<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> die Ungleichung <math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>R<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> erf\u00fcllt ist, was wir gem\u00e4ss Satz <a href=\"..\/..\/chapter\/riemann-summen#x1-180002r47\">6.47<\/a> f\u00fcr gen\u00fcgend kleine Maschenweiten von&nbsp;<math display=\"inline\"><mi>\u2128<\/mi><\/math> erzielen k\u00f6nnen. <\/p><p class=\"indent\">Des Weiteren gilt auch die Dreiecksungleichung (vergleiche zu Satz <a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-113002r24\">4.24<\/a>) f\u00fcr vektorwertige Integrale: F\u00fcr eine stetige Funktion <math display=\"inline\"><mstyle><mi>f<\/mi><\/mstyle> <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> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> ist <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>f<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn> <\/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> Riemann-integrierbar (da stetig) und es gilt <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> \u2225<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mstyle><mi>f<\/mi><\/mstyle><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><msub><mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> \u2225<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>f<\/mi><\/mstyle> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-181001r12\" \/><mstyle class=\"maketag\"><mtext>(6.12)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Die analoge Ungleichung gilt auch f\u00fcr andere Normen. <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z896f311db109\"> <a id=\"x1-181002r51\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 6.51 <\/span>(Dreiecksungleichung f\u00fcr vektorwertige Integrale)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Beweisen Sie Ungleichung <\/span><a href=\"..\/..\/chapter\/riemann-summen#x1-181001r12\"><span class=\"ecti-1095\">6.12<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Verwenden Sie die Tatsache, dass Satz <\/span><a href=\"..\/..\/chapter\/riemann-summen#x1-180002r47\"><span class=\"ecti-1095\">6.47<\/span><\/a> <span class=\"ecti-1095\">auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r vektorwertige Funktionen<\/span> <span class=\"ecti-1095\">g<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ltig ist.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Obige Diskussion enth\u00e4lt mit&nbsp;<math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><\/math> auch den Fall von komplexwertigen Funktionen <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/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\">welche also genau dann Riemann-integrierbar sind, wenn <math display=\"inline\"><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><mi class=\"qopname\">Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> Riemann-integrierbar sind. Das Riemann-Integral ist in diesem Fall gegeben durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi class=\"qopname\"> Re<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/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-bin\">+<\/mo><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi class=\"qopname\"> Im<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/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-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <a id=\"x1-181003r180\"><\/a> \n","rendered":"\n<style scoped=\"scoped\">.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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width:125%;}\ndt {text-align:right; font-weight:bold; clear:left; float:left;}\ndd {width:100%; padding-left:1em; padding-top: 0px; clear:right;}\ndd + dd {float:right; clear:both;}\ndd + dt {clear:both;}\ndt + dt {width: 100%; float: none; padding: 0 70% 0 0;}\ndt + dt + dd {margin-top: -2em;}\ndt + dt + dd + dt {margin-top: 2em;}\n<\/style>\n<style scoped=\"scoped\">\n\/* CSS Analysis-Skript D-Math ETHZ *\/\n\n\/* Uniform Font, also for headers *\/\nh3 {\n\tfont-family: \"Times New Roman\", serif;\n\tmargin-bottom: 35px;\n}\nh4 {\n\tfont-family: \"Times New Roman\", serif;\n}\nh5 {\n\tfont-family: \"Times New Roman\", serif;\n}\n\n\/* Bold font, e.g. for definitions *\/\n.ecbx-1095 {font-weight: 550 ;}\n\n\n\/* Uniform spacing, indent: larger, noindent, enumerate, itemize *\/\np.indent {\n\tmargin: 25px 0px 0px 0px;\n\ttext-indent: 0px; \n}\np.noindent {\n\tmargin: 15px 0px 0px 0px;\n\ttext-indent: 0px; \n}\ndl.enumerate {\n\tmargin: 0px 0px 0px 0px;\n}\ndl.enumerate dt, dl.enumerate dd {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\ndiv.custom-itemize {\n\tmargin: 0px 0px 0px 0px;\n}\ndiv.custom-itemize div.item-head {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n\ttext-align: center;\n}\ndiv.custom-itemize div.item-head:first-of-type {\n\tmargin-top: 0px;\n} \ndiv.custom-itemize div.item-content {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\n.MJXc-display {\n\tmargin: 15px 0px 0px 0px;\n}\n\n\n\n\/* green metheorem\/melemma CSS class for more\/medium important latex-theorem-environments *\/\n\/* metheorem box+header *\/\ndiv.metheorem {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #c7e4da;\n}\ndiv.metheorem h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* melemma box+header *\/\ndiv.melemma {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #F2F2F2;\n}\ndiv.melemma h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* meexample box+header *\/\ndiv.meexample {\n    margin-bottom: 30px;\n    margin-top: 30px;\n\tpadding: 0px 15px 15px 15px;\n\tborder-color: gainsboro;\n\tborder-style: solid;\n\tborder-width: thin;\n}\ndiv.meexample h4 {\n\tfont-size: inherit;\n\tfont-weight: bold;\n    padding: 15px 0px 0px 0px;\n\tmargin-top: 0px;\n\tmargin-bottom: 5px;\n}\ndiv.meexample h4+p.noindent, div.meexample h4+p.indent {\n\tmargin-top: 5px;\n\ttext-indent: 0px;\n}\n\/* padding and margins for stuff inside these boxes, CSS-selector &gt; doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"zde44a1c6de94\" class=\"sectionHead\"><span class=\"titlemark\">6.5 <\/span> <a id=\"x1-1800005\"><\/a>Riemann-Summen<\/h3> <p class=\"noindent\">Riemann gab <math display=\"inline\"><mn>1<\/mn><mn>8<\/mn><mn>5<\/mn><mn>4<\/mn><\/math> eine formale Definition des Integrals mit Hilfe sogenannter Riemann-Summen und eines \u201eGrenz\u00fcbergangs\u201c, dessen Definition \u00c4hnlichkeiten zu den Definitionen des Grenzwertes einer Folge und des Grenzwertes einer Funktion aufweist. Wie in Kapitel <a href=\"..\/..\/part\/das-riemann-integral#x1-1060004\">4<\/a> sei im Folgenden <math display=\"inline\"><mi>f<\/mi><\/math> eine reellwertige Funktion auf einem kompakten Intervall <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z6b4e9e8cd269\"> <a id=\"x1-180001r46\"><\/a> <span class=\"ecbx-1095\">Definition 6.46 <\/span>(Riemann-Summen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">F\u00fcr eine Zerlegung <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> 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> definieren wir die <span class=\"ecbx-1095\">Maschenweite der Zerlegung <\/span><math display=\"inline\"><mi>\u2128<\/mi><\/math> als <span class=\"maperiod\"><math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>\u2128<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><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><\/math><\/span><span class=\"period\">.<\/span> Weiters bezeichnen wir <math display=\"inline\"><mstyle><mi>z<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> als eine <span class=\"ecbx-1095\">erlaubte Wahl von Zwischenpunkten <\/span>der Zerlegung&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi>\u2128<\/mi><\/math><\/span><span class=\"period\">,<\/span> falls <math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi> <\/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> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo> <mo class=\"MathClass-punc\">,<\/mo> <mi>n<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> F\u00fcr eine reellwertige Funktion&nbsp;<math display=\"inline\"><mi>f<\/mi><\/math> auf <span class=\"maperiod\"><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><span class=\"period\">,<\/span> eine Zerlegung&nbsp;<math display=\"inline\"><mi>\u2128<\/mi><\/math> von&nbsp;<math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> und eine erlaubte Wahl von Zwischenpunkten&nbsp;<math display=\"inline\"><mstyle><mi>z<\/mi><\/mstyle><\/math> definieren wir die <span class=\"ecbx-1095\">Riemann-Summe <\/span>durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>R<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>z<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo> <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><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/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-close\">)<\/mo><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\">Das heisst, wir betrachten beliebige Punkte <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi> <\/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> die Funktionswerte <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> an diesen Punkten und hoffen, dass diese halbwegs repr\u00e4sentativ f\u00fcr die Funktionswerte von <math display=\"inline\"><mi>f<\/mi><msub><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><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> <\/mrow> <\/msub> <\/math> sind. Diese Hoffnung mag zwar nicht in allen Teilintervallen immer zutreffen, trotzdem ist die Riemann-Summe eine Approximation des Riemann-Integrals in folgendem Sinne. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z207e86d09555\"> <a id=\"x1-180002r47\"><\/a> <span class=\"ecbx-1095\">Satz 6.47 <\/span>(Riemann-Integral \u00fcber Riemann-Summen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">eine Riemann-integrierbare<\/span> <span class=\"ecti-1095\">reellwertige Funktion auf <\/span><span class=\"maperiod\"><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><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann ist <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">der Grenzwert<\/span> <span class=\"ecti-1095\">der Riemann-Summen <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>R<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">wenn die Maschenweite <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">der Zerlegung gegen Null geht. Formal geschrieben gilt also<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2203<\/mi><mo> <\/mo><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>\u2128<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mstyle><mi>z<\/mi><\/mstyle> <mo class=\"MathClass-punc\">:<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>\u2128<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <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\" \/> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>R<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>z<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>\u2128<\/mi><\/math> <span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber die<\/span> <span class=\"ecti-1095\">Zerlegungen von <\/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\">l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">uft und<\/span> <math display=\"inline\"><mstyle><mi>z<\/mi><\/mstyle><\/math> <span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber die erlaubten Wahlen von<\/span> <span class=\"ecti-1095\">Zwischenpunkten der Zerlegung<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u2128<\/mi><\/math> <span class=\"ecti-1095\">(wie in Definition <\/span><a href=\"..\/..\/chapter\/riemann-summen#x1-180001r46\"><span class=\"ecti-1095\">6.46<\/span><\/a><span class=\"ecti-1095\">) l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">uft.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z8f993e43a7a7\"> <span class=\"ecti-1095\">Bemerkung.<\/span><\/h4> <p class=\"indent\">Die Konvergenz der Riemann-Summen wie in obigem Satz ist sogar eine Charakterisierung der Riemann-Integrierbarkeit \u2013 siehe \u00dcbung&nbsp;<a href=\"..\/..\/chapter\/riemann-summen#x1-180004r49\">6.49<\/a>. Es gibt auch noch weitere, \u00e4quivalente Bedingungen, aber wir begn\u00fcgen uns mit der Aussage in Satz <a href=\"..\/..\/chapter\/riemann-summen#x1-180002r47\">6.47<\/a>. <\/p> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis von Satz <a href=\"..\/..\/chapter\/riemann-summen#x1-180002r47\">6.47<\/a>.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\"> Wir beweisen den Satz zuerst unter der zus\u00e4tzlichen Annahme, dass <math display=\"inline\"><mi>f<\/mi><\/math> stetig ist. Dank dem Sandwich-Kriterium f\u00fcr Riemann-Integrierbarkeit aus Proposition&nbsp;<a href=\"..\/..\/chapter\/integrierbarkeit-stetiger-funktionen#x1-124001r44\">4.44<\/a> wird sich dies als ausreichend herausstellen. <\/p><p class=\"indent\">Sei also <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> stetig 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&nbsp;<a href=\"..\/..\/chapter\/stetige-funktionen-auf-kompakten-intervallen#x1-102005r77\">3.77<\/a> ist <math display=\"inline\"><mi>f<\/mi><\/math> gleichm\u00e4ssig stetig, womit <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> existiert mit <math display=\"inline\"><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><\/math> 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> welche <math display=\"inline\"><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><\/math> erf\u00fcllen. 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 mit Maschenweite <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><\/math> und <math display=\"inline\"><mstyle><mi>z<\/mi><\/mstyle> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> eine erlaubte Wahl von Zwischenpunkten. Dann gilt f\u00fcr alle <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> wegen der Dreiecksungleichung f\u00fcr das Riemann-Integral in Satz&nbsp;<a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-113002r24\">4.24<\/a> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><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><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>z<\/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><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><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><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>z<\/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><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">\u2264<\/mo><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><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><\/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\">&lt;<\/mo> <mi>\ud835\udf00<\/mi> <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-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\">In der Tat is <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/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> <math display=\"inline\"><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-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><\/math> wegen <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><\/math> und somit <math display=\"inline\"><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><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Insbesondere ist <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>R<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>z<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"> <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>1<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><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><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>z<\/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><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">&lt;<\/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><mi>\ud835\udf00<\/mi> <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><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Da <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> die Zerlegung <math display=\"inline\"><mi>\u2128<\/mi><\/math> mit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><\/math> sowie die erlaubten Zwischenpunkte <math display=\"inline\"><mstyle><mi>z<\/mi><\/mstyle><\/math> beliebig waren, beweist dies die Proposition f\u00fcr alle stetigen Funktionen <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Sei nun <math display=\"inline\"><mi>f<\/mi><\/math> eine beliebige Riemann-integrierbare Funktion 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\/integrierbarkeit-stetiger-funktionen#x1-124001r44\">4.44<\/a> existieren <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/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> 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 <span class=\"maperiod\"><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><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><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">.<\/span> Nach der oben schon bewiesenen Aussage sei <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> mit der Eigenschaft, dass f\u00fcr alle Zerlegungen <math display=\"inline\"><mi>\u2128<\/mi><\/math> mit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><\/math> und f\u00fcr alle erlaubten Zwischenpunkte <math display=\"inline\"><mstyle><mi>z<\/mi><\/mstyle><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>R<\/mi> <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-punc\">,<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>z<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <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><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und genauso f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/math><\/span><span class=\"period\">.<\/span> Nun gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>R<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>z<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>R<\/mi> <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-punc\">,<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>z<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <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><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>\ud835\udf00<\/mi><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">&lt;<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>\ud835\udf00<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und unter Verwendung von <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><\/msub><\/math> auf \u00e4hnliche Weise <span class=\"maperiod\"><math display=\"inline\"><mi>R<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>z<\/mi><\/mstyle><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">&gt;<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\ud835\udf00<\/mi><\/math><\/span><span class=\"period\">.<\/span> Damit ist der Satz bewiesen. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z63da9ff1d63a\"> <a id=\"x1-180003r48\"><\/a> <span class=\"ecbx-1095\">Applet 6.48 <\/span>(Riemann-Summen f\u00fcr die Parabel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <div class=\"wp-nocaption \"><\/div><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"532px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/gdcuwv7x\/width\/688\/height\/532\/border\/888888\/rc\/false\/ai\/false\/sdz\/true\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir sehen Riemann-Summen f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r die Parabel aus Abschnitt <\/span><a href=\"..\/..\/chapter\/quadratur-der-parabel#x1-40001\"><span class=\"ecti-1095\">1.1<\/span><\/a><span class=\"ecti-1095\">, wobei die Zwischenpunkte<\/span> <span class=\"ecti-1095\">zuf<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">llig gew<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlt werden.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z5de9c3e879fc\"> <a id=\"x1-180004r49\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 6.49 <\/span>(Charakterisierung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">reellen Zahlen und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie die Umkehrung zu Satz <\/span><a href=\"..\/..\/chapter\/riemann-summen#x1-180002r47\"><span class=\"ecti-1095\">6.47<\/span><\/a><span class=\"ecti-1095\">, also dass die Konvergenz der Riemann-Summen f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">zu einer Zahl <\/span><math display=\"inline\"><mi>I<\/mi><\/math> <span class=\"ecti-1095\">(wie in Satz <\/span><a href=\"..\/..\/chapter\/riemann-summen#x1-180002r47\"><span class=\"ecti-1095\">6.47<\/span><\/a><span class=\"ecti-1095\">) die Riemann-Integrierbarkeit von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">und die Gleichung <\/span><math display=\"inline\"><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>I<\/mi><\/math> <span class=\"ecti-1095\">impliziert.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">W<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlen Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein <\/span><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ein <\/span><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">wie in der Konvergenz der Riemann-Summen. Sei <\/span><math display=\"inline\"><mi>\u2128<\/mi><\/math> <span class=\"ecti-1095\">eine Zerlegung mit Maschenweite <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Betrachten Sie nun das Infimum und das Supremum aller Riemannsummen zu erlaubten<\/span> <span class=\"ecti-1095\">Wahlen von Zwischenpunkten der Zerlegung <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u2128<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">und verkn<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">pfen Sie dies mit einer Untersumme und einer Obersumme.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z431171e7ef4d\"> <a id=\"x1-180005r50\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 6.50 <\/span>(Riemann-Integrierbarkeit stetiger Funktion)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Nach <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/riemann-summen#x1-180004r49\"><span class=\"ecti-1095\">6.49<\/span><\/a> <span class=\"ecti-1095\">h<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tten wir die Konvergenz der Riemann-Summen als Definition f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">Riemann-integrierbare Funktionen verwenden k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen. Zeigen Sie, dass stetige Funktionen<\/span> <span class=\"ecti-1095\">Riemann-integrierbar sind in diesem Sinne (ohne Satz<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/integrierbarkeit-stetiger-funktionen#x1-123001r42\"><span class=\"ecti-1095\">4.42<\/span><\/a> <span class=\"ecti-1095\">zu verwenden).<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Hinweis: Betrachten Sie hierzu zuerst eine Folge von Zerlegungen <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u2128<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">mit der Eigenschaft, dass <\/span><math display=\"inline\"><msub><mrow><mi>\u2128<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">feiner ist als <\/span><math display=\"inline\"><msub><mrow><mi>\u2128<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Da uns der Grenzwert einer Folge von Riemann-Summen a priori nicht bekannt ist, kann man<\/span> <span class=\"ecti-1095\">eher zeigen, dass die Folge eine Cauchy-Folge ist.<\/span> <\/p> <\/div> <a id=\"x1-180006r179\"><\/a> <h4 id=\"z64d40a3d21e2\" class=\"subsectionHead\"><span class=\"titlemark\">6.5.1 <\/span> <a id=\"x1-1810001\"><\/a>Vektorwertige Integrale<\/h4> <p class=\"noindent\">Unsere urspr\u00fcngliche Definition des Riemann-Integrals in Kapitel <a href=\"..\/..\/part\/das-riemann-integral#x1-1060004\">4<\/a> verwendete die Ungleichung <math display=\"inline\"><mo class=\"MathClass-rel\">\u2264<\/mo><\/math> in                                                                                                                                                                           <math display=\"inline\"><mi>\u211d<\/mi><\/math> in zentraler Weise und kann deswegen nicht auf diese Weise f\u00fcr vektorwertige Funktionen verallgemeinert werden. Riemann-Summen lassen sich hingegen leicht verallgemeinern. F\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mstyle><mi>f<\/mi><\/mstyle> <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> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <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> und <math display=\"inline\"><mstyle><mi>z<\/mi><\/mstyle> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> eine zul\u00e4ssige Wahl von Zwischenpunkten gem\u00e4ss Definition <a href=\"..\/..\/chapter\/riemann-summen#x1-180001r46\">6.46<\/a> setzen wir wie zuvor <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>R<\/mi><mo class=\"MathClass-open\">(<\/mo><mstyle><mi>f<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-punc\">,<\/mo><mstyle><mi>z<\/mi><\/mstyle><mo class=\"MathClass-close\">)<\/mo> <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><mstyle><mi>f<\/mi><\/mstyle><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow> <mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/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-close\">)<\/mo><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\">Des Weiteren sagen wir, dass <math display=\"inline\"><mstyle><mi>f<\/mi><\/mstyle> <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> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> Riemann-integrierbar ist, falls <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mi>f<\/mi><\/mstyle> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <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> und die Komponentenfunktionen <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>j<\/mi><\/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> f\u00fcr <math display=\"inline\"><mi>j<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><\/math> Riemann-integrierbar sind (man vergleiche dies zu Proposition&nbsp;<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-148002r44\">5.44<\/a>). Das Riemann-Integral wird dann komponentenweise definiert durch                                                                                                                                                                           <\/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><mstyle><mi>f<\/mi><\/mstyle> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mo>\u2026<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mo class=\"MathClass-punc\">,<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow> <mi>d<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup><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\">Satz <a href=\"..\/..\/chapter\/riemann-summen#x1-180002r47\">6.47<\/a> gilt nun analog f\u00fcr Riemann-integrierbare Funktionen 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> nach <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math><\/span><span class=\"period\">.<\/span> In der Tat gilt <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mstyle><mi>f<\/mi><\/mstyle> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>R<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mstyle><mi>f<\/mi><\/mstyle><mo class=\"MathClass-punc\">,<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> genau dann, wenn f\u00fcr alle <math display=\"inline\"><mi>j<\/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>d<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> die Ungleichung <math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msubsup><mrow><mi class=\"MathClass-op\">\u222b  <\/mi><mo> <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>R<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi>\u2128<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> erf\u00fcllt ist, was wir gem\u00e4ss Satz <a href=\"..\/..\/chapter\/riemann-summen#x1-180002r47\">6.47<\/a> f\u00fcr gen\u00fcgend kleine Maschenweiten von&nbsp;<math display=\"inline\"><mi>\u2128<\/mi><\/math> erzielen k\u00f6nnen. <\/p><p class=\"indent\">Des Weiteren gilt auch die Dreiecksungleichung (vergleiche zu Satz <a href=\"..\/..\/chapter\/erste-integrationsgesetze#x1-113002r24\">4.24<\/a>) f\u00fcr vektorwertige Integrale: F\u00fcr eine stetige Funktion <math display=\"inline\"><mstyle><mi>f<\/mi><\/mstyle> <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> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> ist <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>f<\/mi><\/mstyle><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mn>2<\/mn> <\/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> Riemann-integrierbar (da stetig) und es gilt <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> \u2225<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mstyle><mi>f<\/mi><\/mstyle><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><msub><mrow><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> \u2225<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mo class=\"MathClass-rel\">\u2225<\/mo><mstyle><mi>f<\/mi><\/mstyle> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow> <mn>2<\/mn><\/mrow><\/msub><mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-181001r12\" \/><mstyle class=\"maketag\"><mtext>(6.12)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Die analoge Ungleichung gilt auch f\u00fcr andere Normen. <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z896f311db109\"> <a id=\"x1-181002r51\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 6.51 <\/span>(Dreiecksungleichung f\u00fcr vektorwertige Integrale)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Beweisen Sie Ungleichung <\/span><a href=\"..\/..\/chapter\/riemann-summen#x1-181001r12\"><span class=\"ecti-1095\">6.12<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Verwenden Sie die Tatsache, dass Satz <\/span><a href=\"..\/..\/chapter\/riemann-summen#x1-180002r47\"><span class=\"ecti-1095\">6.47<\/span><\/a> <span class=\"ecti-1095\">auch f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r vektorwertige Funktionen<\/span> <span class=\"ecti-1095\">g<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ltig ist.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Obige Diskussion enth\u00e4lt mit&nbsp;<math display=\"inline\"><mi>d<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><\/math> auch den Fall von komplexwertigen Funktionen <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/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\">welche also genau dann Riemann-integrierbar sind, wenn <math display=\"inline\"><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><mi class=\"qopname\">Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> Riemann-integrierbar sind. Das Riemann-Integral ist in diesem Fall gegeben durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mo> \u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi class=\"qopname\"> Re<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/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-bin\">+<\/mo><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><msubsup><mrow><mo>\u222b  <\/mo><\/mrow><mrow><mi>a<\/mi><\/mrow><mrow><mi>b<\/mi><\/mrow><\/msubsup><mi class=\"qopname\"> Im<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/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-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <a id=\"x1-181003r180\"><\/a> \n","protected":false},"author":1089,"menu_order":5,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-72","chapter","type-chapter","status-publish","hentry"],"part":67,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/72","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/users\/1089"}],"version-history":[{"count":0,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/72\/revisions"}],"part":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/parts\/67"}],"metadata":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/72\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=72"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=72"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=72"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=72"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}