{"id":65,"date":"2021-12-15T09:53:12","date_gmt":"2021-12-15T09:53:12","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/stetigkeit-2\/"},"modified":"2021-12-15T09:53:12","modified_gmt":"2021-12-15T09:53:12","slug":"stetigkeit-2","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/stetigkeit-2\/","title":{"raw":"Stetigkeit","rendered":"Stetigkeit"},"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=\"zd319f39be2bf\" class=\"sectionHead\"><span class=\"titlemark\">5.4 <\/span> <a id=\"x1-1490004\"><\/a>Stetigkeit<\/h3> <a id=\"x1-149001r148\"><\/a> <h4 id=\"z853c8fa6ec83\" class=\"subsectionHead\"><span class=\"titlemark\">5.4.1 <\/span> <a id=\"x1-1500001\"><\/a>Definition und Charakterisierungen<\/h4> <p class=\"noindent\">Wir m\u00f6chten nun den Stetigkeitsbegriff auf metrische R\u00e4ume verallgemeinern. Wie wir sehen werden, l\u00e4sst sich dieser auch ausschliesslich mittels Folgen charakterisieren. Wir definieren zuerst zwei a priori verschiedene Begriffe in folgender Definition und zeigen danach, dass diese Begriffe doch gleich sind. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z76931d1164b6\"> <a id=\"x1-150001r48\"><\/a> <span class=\"ecbx-1095\">Definition 5.48 <\/span>(Stetigkeit bei einem Punkt)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Seien <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>X<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>Y<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>Y<\/mi> <\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> zwei metrische R\u00e4ume und sei <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> eine Funktion. Wir sagen, dass <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecbx-1095\">bei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <math display=\"inline\"><mstyle><mi>\ud835\udf00<\/mi><\/mstyle><\/math><span class=\"ecbx-1095\">-<\/span><math display=\"inline\"><mstyle><mi>\u03b4<\/mi><\/mstyle><\/math><span class=\"ecbx-1095\">-stetig<\/span> ist, falls f\u00fcr alle <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ein <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> existiert, so dass f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\u03b4<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> auch <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> gilt. <\/p><p class=\"indent\">Wir                                                  sagen,                                                  dass <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecbx-1095\">bei<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecbx-1095\">folgenstetig         <\/span>ist,          falls          f\u00fcr          jede          konvergente          Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> in <math display=\"inline\"><mi>X<\/mi><\/math> mit                                                                                                        Grenzwert <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> die                                                                                                               Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> konvergiert                                            und                                            Grenzwert <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> hat.                                                                                                                                                                           <\/p> <\/div> <p class=\"indent\">Wie in obiger Definition und vielleicht schon in den letzten zwei Abschnitten ersichtlich wurde, ist eine explizite Bezeichnung <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/math> der Metrik in <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> nicht immer notwendig (da man zum Beispiel oft stattdessen nur B\u00e4lle betrachtet) und f\u00fchrt nur zu zus\u00e4tzlicher Notation. Wir werden deswegen in Zukunft schlicht <math display=\"inline\"><mi>X<\/mi><\/math> als metrischen Raum bezeichnen, wobei die Metrik also implizit ist, keinen \u201eNamen\u201c hat und wenn n\u00f6tig einfach mit <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> bezeichnet wird. <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z7125266cd445\"> <a id=\"x1-150002r49\"><\/a> <span class=\"ecbx-1095\">Lemma 5.49 <\/span>(Stetigkeit bei einem Punkt)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">zwei metrische R<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ume, <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">eine Funktion und <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">ein Punkt. Dann ist <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">genau dann bei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math><span class=\"ecti-1095\">-<\/span><math display=\"inline\"><mi>\u03b4<\/mi><\/math><span class=\"ecti-1095\">-stetig,<\/span> <span class=\"ecti-1095\">wenn <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">folgenstetig ist.<\/span> <\/p> <\/div> <p class=\"indent\">Auf Grund der Aussage des obigen Lemma sagen wir auch kurz, dass <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecbx-1095\">bei<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub><\/math> <span class=\"ecbx-1095\">stetig <\/span>ist, falls <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math>-<math display=\"inline\"><mi>\u03b4<\/mi><\/math>-stetig ist. Des Weiteren ist <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecbx-1095\">stetig<\/span>, wenn <math display=\"inline\"><mi>f<\/mi><\/math> bei jedem Punkt in <math display=\"inline\"><mi>X<\/mi><\/math> stetig ist. In Proposition <a href=\"..\/..\/chapter\/stetigkeit#x1-150003r50\">5.50<\/a> werden weitere wichtige Charakterisierungen von Stetigkeit folgen. <\/p><p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Angenommen <math display=\"inline\"><mi>f<\/mi><\/math> ist bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math>-<math display=\"inline\"><mi>\u03b4<\/mi><\/math>-stetig. Sei <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> eine Folge in <math display=\"inline\"><mi>X<\/mi><\/math> mit Grenzwert <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> 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> Dann existiert ein <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> so dass <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\u03b4<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Da <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> gegen <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> konvergiert, existiert nun ein <math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\u03b4<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> Insbesondere gilt f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> also <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Da <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig war, gilt somit <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und <math display=\"inline\"><mi>f<\/mi><\/math> ist bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> folgenstetig wie gew\u00fcnscht. <\/p><p class=\"indent\">Angenommen <math display=\"inline\"><mi>f<\/mi><\/math> ist bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> nicht <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math>-<math display=\"inline\"><mi>\u03b4<\/mi><\/math>-stetig. Dann existiert ein <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> so dass es f\u00fcr jedes <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ein <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\u03b4<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> gibt mit <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2209<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Wir w\u00e4hlen nun f\u00fcr jedes <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><\/math> ein solches <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Die Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> konvergiert somit gegen <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> und es gilt <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2209<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> Insbesondere konvergiert <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> nicht gegen <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><mi>f<\/mi><\/math> ist nicht folgenstetig. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Wir wollen nun die a priori verschiedenen Begriffe der Stetigkeit zueinander in Beziehung bringen. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z56f468a3dc70\"> <a id=\"x1-150003r50\"><\/a> <span class=\"ecbx-1095\">Proposition 5.50 <\/span>(Charakterisierungen der Stetigkeit)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">zwei<\/span> <span class=\"ecti-1095\">metrische R<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ume und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">eine Funktion. Dann sind folgende Bedingungen <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent:<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Funktion <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">ist stetig.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><mi>x<\/mi><\/math> <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math><span class=\"ecti-1095\">-<\/span><math display=\"inline\"><mi>\u03b4<\/mi><\/math><span class=\"ecti-1095\">-stetig.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><mi>x<\/mi><\/math> <span class=\"ecti-1095\">folgenstetig.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">und f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede Umgebung <\/span><math display=\"inline\"><mi>U<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>U<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">eine Umgebung von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <\/div> <p class=\"indent\">\ud83e\ude86Die einzelnen Implikationen sind Matrjoschka-Beweise. <\/p><p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Per Definition der Stetigkeit und Lemma <a href=\"..\/..\/chapter\/stetigkeit#x1-150002r49\">5.49<\/a> sind (i), (ii) und (iii) \u00e4quivalent. Wir zeigen als n\u00e4chsten Schritt die \u00c4quivalenz von (i) und (iv).                                                                                                                                                                           <\/p><p class=\"indent\">Sei <math display=\"inline\"><mi>f<\/mi><\/math> stetig, sei <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> und sei <math display=\"inline\"><mi>U<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>Y<\/mi> <\/math> eine Umgebung von <span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Per Definition des Umgebungsbegriffes gibt es ein <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>U<\/mi><\/math><\/span><span class=\"period\">.<\/span> Da <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><mi>x<\/mi><\/math> <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math>-<math display=\"inline\"><mi>\u03b4<\/mi><\/math>-stetig ist, gibt es ein <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> mit <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\u03b4<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>U<\/mi><\/math> und somit <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\u03b4<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>U<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Also ist <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>U<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> eine Umgebung von <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span> was die Implikation (i)<math display=\"inline\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><\/math>(iv) beweist. <\/p><p class=\"indent\">Angenommen <math display=\"inline\"><mi>f<\/mi><\/math> erf\u00fcllt die Bedingung in (iv). Sei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Wir wissen, dass <math display=\"inline\"><mi>U<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> eine Umgebung von <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math> ist. Also ist <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> eine Umgebung von <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> womit per Definition ein <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> existiert mit <math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\u03b4<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> oder \u00e4quivalenterweise <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\u03b4<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Also ist <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math>-<math display=\"inline\"><mi>\u03b4<\/mi><\/math>-stetig, was die Implikation (iv)<math display=\"inline\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><\/math>(i) beweist. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Nach Proposition <a href=\"..\/..\/chapter\/stetigkeit#x1-150003r50\">5.50<\/a> verf\u00fcgen wir nun \u00fcber zwei Varianten, wie wir Stetigkeit (oder Nicht-Stetigkeit) einer Funktion nachweisen k\u00f6nnen: mit der Definition (dem <math display=\"inline\"><mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u03b4<\/mi><\/math>-Spiel) oder mit Konvergenz von Folgen. Letztere Variante kann unter anderem sehr n\u00fctzlich sein, wenn man Nicht-Stetigkeit einer Funktion zeigen will, da man demnach bloss eine spezielle Folge konstruieren muss, die auf eine nicht-konvergente Folge abgebildet wird. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zeece4707829a\"> <a id=\"x1-150008r51\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.51.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">eine Teilmenge und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">eine stetige Funktion. Angenommen <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/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\">ist eine Folge in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergiert. Muss auch <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/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\">konvergieren?<\/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\">Betrachten Sie die einfachste stetige Funktion aus Beispiel <\/span><a href=\"..\/..\/chapter\/stetigkeit#x1-94005r47\"><span class=\"ecti-1095\">3.47<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/p><\/details>  <\/div> <a id=\"x1-150009r150\"><\/a> <h4 id=\"z494b5c04843e\" class=\"subsectionHead\"><span class=\"titlemark\">5.4.2 <\/span> <a id=\"x1-1510002\"><\/a>Zwei st\u00e4rkere Stetigkeitsbegriffe<\/h4> <p class=\"noindent\">Wie wir bereits gesehen haben, sind manchmal folgende Stetigkeitseigenschaften n\u00fctzlich. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zf612ddcaf437\"> <a id=\"x1-151001r52\"><\/a> <span class=\"ecbx-1095\">Definition 5.52.<\/span> <\/h4> <p class=\"indent\">Seien <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>X<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>Y<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo>  <\/mrow><mrow><mi>Y<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math> zwei metrische R\u00e4ume und <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> eine Funktion. Dann heisst <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecbx-1095\">gleichm<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">ssig<\/span> <span class=\"ecbx-1095\">stetig<\/span>, falls es zu jedem <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ein <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> gibt, so dass f\u00fcr alle <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> mit <math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo>  <\/mrow><mrow><mi>X<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><\/math> auch <math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>Y<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> gilt. Des Weiteren heisst <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecbx-1095\">Lipschitz-stetig<\/span>, falls es eine sogenannte <span class=\"ecbx-1095\">Lipschitz-Konstante<\/span> <math display=\"inline\"><mi>L<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> gibt mit                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>Y<\/mi> <\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>L<\/mi><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>X<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Per Definition sind gleichm\u00e4ssig stetige Funktionen stetig und wie wir schon gesehen haben, sind stetige Funktionen nicht zwingend gleichm\u00e4ssig stetig. Des Weiteren sind Lipschitz stetige Funktionen gleichm\u00e4ssig stetig. (Wieso?<button class=\"hover-trigger\">(Wieso?)<\/button><span class=\"hover-text\"><span class=\"marginpar\">Wir k\u00f6nnen f\u00fcr jedes <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> einfach <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>L<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><mi>\ud835\udf00<\/mi><\/math> verwenden. (Wobei wir <math display=\"inline\"><mi>L<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math> im Z\u00e4hler verwenden um auch im eigenartigen Fall <math display=\"inline\"><mi>L<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> eine vern\u00fcnftige Definition f\u00fcr <math display=\"inline\"><mi>\u03b4<\/mi><\/math> zu erhalten.)<\/span><\/span>). Unter gewissen Bedingungen an den metrischen Raum <math display=\"inline\"><mi>X<\/mi><\/math> sind stetige Funktionen gleichm\u00e4ssig stetig (zum Beispiel falls <math display=\"inline\"><mi>X<\/mi><\/math> ein kompaktes Intervall ist); wir werden im zweiten Semester darauf zur\u00fcckkommen. <a id=\"x1-151002r151\"><\/a> <\/p> <h4 id=\"z0ff499c5ef5a\" class=\"subsectionHead\"><span class=\"titlemark\">5.4.3 <\/span> <a id=\"x1-1520003\"><\/a>Konstruktion von stetigen Funktionen<\/h4> <p class=\"noindent\">Wie schon in Abschnitt <a href=\"..\/..\/chapter\/stetigkeit#x1-940005\">3.5<\/a> lassen sich f\u00fcr stetigen Funktionen verschiedene Operationen durchf\u00fchren, die wiederum zu weiteren stetigen Funktionen f\u00fchren. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z58ed13307753\"> <a id=\"x1-152001r53\"><\/a> <span class=\"ecbx-1095\">Proposition 5.53 <\/span>(Stetige Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>Y<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>Z<\/mi><\/math> <span class=\"ecti-1095\">metrische R<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ume.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>Y<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Z<\/mi><\/math> <span class=\"ecti-1095\">stetig sind, dann ist auch <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Z<\/mi><\/math> <span class=\"ecti-1095\">stetig.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Eine Funktion <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">ist genau dann stetig, wenn die Komponenten<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><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> <span class=\"ecti-1095\">stetig sind,<\/span> <span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">:<\/mo> <msup><mrow><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>d<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><mo class=\"MathClass-rel\">\u21a6<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <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> <span class=\"ecti-1095\">die Projektion<\/span> <span class=\"ecti-1095\">auf die <\/span><math display=\"inline\"><mi>j<\/mi><\/math><span class=\"ecti-1095\">-te<\/span> <span class=\"ecti-1095\">Koordinate bezeichnet.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Addition <\/span><math display=\"inline\"><mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">ist stetig.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Multiplikation <\/span><math display=\"inline\"><mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">ist stetig.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(v)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Kehrwertfunktion <\/span><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-punc\">:<\/mo> <mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u00d7<\/mo><\/mrow><\/msup><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>z<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">ist stetig.<\/span><\/dd><\/dl> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir verwenden im Beweis mehrmals die Charakterisierung von Stetigkeit mittels Folgen aus Proposition&nbsp;<a href=\"..\/..\/chapter\/stetigkeit#x1-150003r50\">5.50<\/a>.                                                                                                                                                                           <\/p><p class=\"indent\">F\u00fcr (i) sei <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> eine konvergente Folge in <math display=\"inline\"><mi>X<\/mi><\/math> mit Grenzwert <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> Da <math display=\"inline\"><mi>f<\/mi><\/math> stetig ist, konvergiert die Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> (nach Proposition&nbsp;<a href=\"..\/..\/chapter\/stetigkeit#x1-150003r50\">5.50<\/a>) gegen <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Nach Stetigkeit von <math display=\"inline\"><mi>g<\/mi><\/math> konvergiert <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> gegen <span class=\"maperiod\"><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Dies aber impliziert die Aussage in (i) wegen Proposition&nbsp;<a href=\"..\/..\/chapter\/stetigkeit#x1-150003r50\">5.50<\/a>. <\/p><p class=\"indent\">F\u00fcr (ii) bemerken wir zuerst, dass die Projektion <math display=\"inline\"><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">:<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> wegen <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>v<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mi>v<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>w<\/mi><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/math> f\u00fcr <math display=\"inline\"><mi>v<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> Lipschitz-stetig mit Lipschitz-Konstante <math display=\"inline\"><mn>1<\/mn><\/math> ist (bez\u00fcglich der Metrik auf <math display=\"inline\"><msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> induziert durch <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/math>). Daher folgt aus (i) und Stetigkeit von <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> dass auch <math display=\"inline\"><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>C<\/mi><\/math> stetig ist f\u00fcr jedes <span class=\"maperiod\"><math display=\"inline\"><mi>j<\/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>d<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">.<\/span> Sei nun umgekehrt <math display=\"inline\"><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> f\u00fcr jedes <math display=\"inline\"><mi>j<\/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>d<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> stetig. Sei <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> eine konvergente Folge in <math display=\"inline\"><mi>X<\/mi><\/math> mit Grenzwert <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> Also konvergiert <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> f\u00fcr alle <math display=\"inline\"><mi>j<\/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>d<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> nach <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> was nach Proposition&nbsp;<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-148002r44\">5.44<\/a> impliziert, dass <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> nach <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> konvergiert. Dies beweist (ii). <\/p><p class=\"indent\">F\u00fcr <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> gilt <\/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><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>w<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>2<\/mn><mo class=\"MathClass-rel\">\u2225<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2225<\/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\">Daher ist die Addition Lipschitz-stetig mit Lipschitz-Konstante <span class=\"maperiod\"><math display=\"inline\"><mn>2<\/mn><\/math><\/span><span class=\"period\">,<\/span> und (iii) folgt. <\/p><p class=\"indent\">F\u00fcr <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mi>w<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mi>w<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mi>w<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/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\">Sei nun <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> min<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi>\ud835\udf00<\/mi><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dann gilt <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/math> und daher                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mi>w<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dies zeigt Stetigkeit der Multiplikation in (iv). <\/p><p class=\"indent\">F\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u00d7<\/mo><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u00d7<\/mo> <\/mrow> <\/msup> <\/math> mit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> min<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>\ud835\udf00<\/mi><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> gilt <\/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>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2265<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&gt;<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und daher <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>z<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msubsup><mrow><mi>z<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><msub><mrow><mi>z<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>2<\/mn><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/mrow><\/msup><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi>\ud835\udf00<\/mi><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Da <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig war, zeigt dies die Stetigkeit der Kehrwertabbildung in (v). <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z20d04ce420cf\"> <a id=\"x1-152007r54\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.54 <\/span>(Distanzfunktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein metrischer Raum. In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir den Abstand von Teilmengen von<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">zu Punkten<\/span> <span class=\"ecti-1095\">diskutieren. Zu <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">nicht-leer definieren wir<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>a<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi>A<\/mi><\/mrow><\/munder><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Zeigen Sie, dass die Funktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">Lipschitz-stetig mit Lipschitz-Konstante <\/span><math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">ist.<\/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\">Sie k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/metrische-raeume#x1-140009r14\"><span class=\"ecti-1095\">5.14<\/span><\/a> <span class=\"ecti-1095\">verwenden.<\/span><\/p><\/details>  <\/div> <a id=\"x1-152008r149\"><\/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; 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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=\"zd319f39be2bf\" class=\"sectionHead\"><span class=\"titlemark\">5.4 <\/span> <a id=\"x1-1490004\"><\/a>Stetigkeit<\/h3> <a id=\"x1-149001r148\"><\/a> <h4 id=\"z853c8fa6ec83\" class=\"subsectionHead\"><span class=\"titlemark\">5.4.1 <\/span> <a id=\"x1-1500001\"><\/a>Definition und Charakterisierungen<\/h4> <p class=\"noindent\">Wir m\u00f6chten nun den Stetigkeitsbegriff auf metrische R\u00e4ume verallgemeinern. Wie wir sehen werden, l\u00e4sst sich dieser auch ausschliesslich mittels Folgen charakterisieren. Wir definieren zuerst zwei a priori verschiedene Begriffe in folgender Definition und zeigen danach, dass diese Begriffe doch gleich sind. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z76931d1164b6\"> <a id=\"x1-150001r48\"><\/a> <span class=\"ecbx-1095\">Definition 5.48 <\/span>(Stetigkeit bei einem Punkt)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Seien <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>X<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>Y<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>Y<\/mi> <\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> zwei metrische R\u00e4ume und sei <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> eine Funktion. Wir sagen, dass <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecbx-1095\">bei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <math display=\"inline\"><mstyle><mi>\ud835\udf00<\/mi><\/mstyle><\/math><span class=\"ecbx-1095\">&#8211;<\/span><math display=\"inline\"><mstyle><mi>\u03b4<\/mi><\/mstyle><\/math><span class=\"ecbx-1095\">-stetig<\/span> ist, falls f\u00fcr alle <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ein <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> existiert, so dass f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\u03b4<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> auch <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> gilt. <\/p><p class=\"indent\">Wir                                                  sagen,                                                  dass <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecbx-1095\">bei<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecbx-1095\">folgenstetig         <\/span>ist,          falls          f\u00fcr          jede          konvergente          Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> in <math display=\"inline\"><mi>X<\/mi><\/math> mit                                                                                                        Grenzwert <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> die                                                                                                               Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> konvergiert                                            und                                            Grenzwert <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> hat.                                                                                                                                                                           <\/p> <\/div> <p class=\"indent\">Wie in obiger Definition und vielleicht schon in den letzten zwei Abschnitten ersichtlich wurde, ist eine explizite Bezeichnung <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/math> der Metrik in <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> nicht immer notwendig (da man zum Beispiel oft stattdessen nur B\u00e4lle betrachtet) und f\u00fchrt nur zu zus\u00e4tzlicher Notation. Wir werden deswegen in Zukunft schlicht <math display=\"inline\"><mi>X<\/mi><\/math> als metrischen Raum bezeichnen, wobei die Metrik also implizit ist, keinen \u201eNamen\u201c hat und wenn n\u00f6tig einfach mit <math display=\"inline\"><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-punc\">,<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> bezeichnet wird. <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z7125266cd445\"> <a id=\"x1-150002r49\"><\/a> <span class=\"ecbx-1095\">Lemma 5.49 <\/span>(Stetigkeit bei einem Punkt)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">zwei metrische R<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ume, <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">eine Funktion und <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">ein Punkt. Dann ist <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">genau dann bei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math><span class=\"ecti-1095\">&#8211;<\/span><math display=\"inline\"><mi>\u03b4<\/mi><\/math><span class=\"ecti-1095\">-stetig,<\/span> <span class=\"ecti-1095\">wenn <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">folgenstetig ist.<\/span> <\/p> <\/div> <p class=\"indent\">Auf Grund der Aussage des obigen Lemma sagen wir auch kurz, dass <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecbx-1095\">bei<\/span> <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub><\/math> <span class=\"ecbx-1095\">stetig <\/span>ist, falls <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math>&#8211;<math display=\"inline\"><mi>\u03b4<\/mi><\/math>-stetig ist. Des Weiteren ist <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecbx-1095\">stetig<\/span>, wenn <math display=\"inline\"><mi>f<\/mi><\/math> bei jedem Punkt in <math display=\"inline\"><mi>X<\/mi><\/math> stetig ist. In Proposition <a href=\"..\/..\/chapter\/stetigkeit#x1-150003r50\">5.50<\/a> werden weitere wichtige Charakterisierungen von Stetigkeit folgen. <\/p><div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Angenommen <math display=\"inline\"><mi>f<\/mi><\/math> ist bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math>&#8211;<math display=\"inline\"><mi>\u03b4<\/mi><\/math>-stetig. Sei <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> eine Folge in <math display=\"inline\"><mi>X<\/mi><\/math> mit Grenzwert <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> 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> Dann existiert ein <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> so dass <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\u03b4<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Da <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> gegen <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> konvergiert, existiert nun ein <math display=\"inline\"><mi>N<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> mit <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\u03b4<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math><\/span><span class=\"period\">.<\/span> Insbesondere gilt f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>N<\/mi><\/math> also <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Da <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig war, gilt somit <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und <math display=\"inline\"><mi>f<\/mi><\/math> ist bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> folgenstetig wie gew\u00fcnscht. <\/p><p class=\"indent\">Angenommen <math display=\"inline\"><mi>f<\/mi><\/math> ist bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> nicht <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math>&#8211;<math display=\"inline\"><mi>\u03b4<\/mi><\/math>-stetig. Dann existiert ein <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> so dass es f\u00fcr jedes <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ein <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\u03b4<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> gibt mit <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2209<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Wir w\u00e4hlen nun f\u00fcr jedes <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac><\/math> ein solches <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>n<\/mi><\/mrow><\/mfrac> <\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Die Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> konvergiert somit gegen <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> und es gilt <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2209<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> Insbesondere konvergiert <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> nicht gegen <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><mi>f<\/mi><\/math> ist nicht folgenstetig. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Wir wollen nun die a priori verschiedenen Begriffe der Stetigkeit zueinander in Beziehung bringen. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z56f468a3dc70\"> <a id=\"x1-150003r50\"><\/a> <span class=\"ecbx-1095\">Proposition 5.50 <\/span>(Charakterisierungen der Stetigkeit)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">zwei<\/span> <span class=\"ecti-1095\">metrische R<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ume und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">eine Funktion. Dann sind folgende Bedingungen <\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent:<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Funktion <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">ist stetig.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><mi>x<\/mi><\/math> <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math><span class=\"ecti-1095\">&#8211;<\/span><math display=\"inline\"><mi>\u03b4<\/mi><\/math><span class=\"ecti-1095\">-stetig.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><mi>x<\/mi><\/math> <span class=\"ecti-1095\">folgenstetig.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">und f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede Umgebung <\/span><math display=\"inline\"><mi>U<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>U<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">eine Umgebung von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <\/div> <p class=\"indent\">\ud83e\ude86Die einzelnen Implikationen sind Matrjoschka-Beweise. <\/p><div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Per Definition der Stetigkeit und Lemma <a href=\"..\/..\/chapter\/stetigkeit#x1-150002r49\">5.49<\/a> sind (i), (ii) und (iii) \u00e4quivalent. Wir zeigen als n\u00e4chsten Schritt die \u00c4quivalenz von (i) und (iv).                                                                                                                                                                           <\/p><p class=\"indent\">Sei <math display=\"inline\"><mi>f<\/mi><\/math> stetig, sei <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> und sei <math display=\"inline\"><mi>U<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>Y<\/mi> <\/math> eine Umgebung von <span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Per Definition des Umgebungsbegriffes gibt es ein <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> mit <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>U<\/mi><\/math><\/span><span class=\"period\">.<\/span> Da <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><mi>x<\/mi><\/math> <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math>&#8211;<math display=\"inline\"><mi>\u03b4<\/mi><\/math>-stetig ist, gibt es ein <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> mit <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\u03b4<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>U<\/mi><\/math> und somit <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\u03b4<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>U<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Also ist <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>U<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> eine Umgebung von <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span> was die Implikation (i)<math display=\"inline\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><\/math>(iv) beweist. <\/p><p class=\"indent\">Angenommen <math display=\"inline\"><mi>f<\/mi><\/math> erf\u00fcllt die Bedingung in (iv). Sei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Wir wissen, dass <math display=\"inline\"><mi>U<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> eine Umgebung von <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math> ist. Also ist <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> eine Umgebung von <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> womit per Definition ein <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> existiert mit <math display=\"inline\"><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\u03b4<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> oder \u00e4quivalenterweise <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\u03b4<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <msub><mrow><mi>B<\/mi><\/mrow><mrow><mi>\ud835\udf00<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Also ist <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <math display=\"inline\"><mi>\ud835\udf00<\/mi><\/math>&#8211;<math display=\"inline\"><mi>\u03b4<\/mi><\/math>-stetig, was die Implikation (iv)<math display=\"inline\"><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><\/math>(i) beweist. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Nach Proposition <a href=\"..\/..\/chapter\/stetigkeit#x1-150003r50\">5.50<\/a> verf\u00fcgen wir nun \u00fcber zwei Varianten, wie wir Stetigkeit (oder Nicht-Stetigkeit) einer Funktion nachweisen k\u00f6nnen: mit der Definition (dem <math display=\"inline\"><mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u03b4<\/mi><\/math>-Spiel) oder mit Konvergenz von Folgen. Letztere Variante kann unter anderem sehr n\u00fctzlich sein, wenn man Nicht-Stetigkeit einer Funktion zeigen will, da man demnach bloss eine spezielle Folge konstruieren muss, die auf eine nicht-konvergente Folge abgebildet wird. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zeece4707829a\"> <a id=\"x1-150008r51\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.51.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">eine Teilmenge und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">eine stetige Funktion. Angenommen <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/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\">ist eine Folge in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">konvergiert. Muss auch <\/span><math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>a<\/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\">konvergieren?<\/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\">Betrachten Sie die einfachste stetige Funktion aus Beispiel <\/span><a href=\"..\/..\/chapter\/stetigkeit#x1-94005r47\"><span class=\"ecti-1095\">3.47<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/p><\/details>  <\/div> <a id=\"x1-150009r150\"><\/a> <h4 id=\"z494b5c04843e\" class=\"subsectionHead\"><span class=\"titlemark\">5.4.2 <\/span> <a id=\"x1-1510002\"><\/a>Zwei st\u00e4rkere Stetigkeitsbegriffe<\/h4> <p class=\"noindent\">Wie wir bereits gesehen haben, sind manchmal folgende Stetigkeitseigenschaften n\u00fctzlich. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zf612ddcaf437\"> <a id=\"x1-151001r52\"><\/a> <span class=\"ecbx-1095\">Definition 5.52.<\/span> <\/h4> <p class=\"indent\">Seien <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>X<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>Y<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo>  <\/mrow><mrow><mi>Y<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math> zwei metrische R\u00e4ume und <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> eine Funktion. Dann heisst <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecbx-1095\">gleichm<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">ssig<\/span> <span class=\"ecbx-1095\">stetig<\/span>, falls es zu jedem <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ein <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> gibt, so dass f\u00fcr alle <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> mit <math display=\"inline\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo>  <\/mrow><mrow><mi>X<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><\/math> auch <math display=\"inline\"><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>Y<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><\/math> gilt. Des Weiteren heisst <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecbx-1095\">Lipschitz-stetig<\/span>, falls es eine sogenannte <span class=\"ecbx-1095\">Lipschitz-Konstante<\/span> <math display=\"inline\"><mi>L<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> gibt mit                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>Y<\/mi> <\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>L<\/mi><msub><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mi>X<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Per Definition sind gleichm\u00e4ssig stetige Funktionen stetig und wie wir schon gesehen haben, sind stetige Funktionen nicht zwingend gleichm\u00e4ssig stetig. Des Weiteren sind Lipschitz stetige Funktionen gleichm\u00e4ssig stetig. (Wieso?<button class=\"hover-trigger\">(Wieso?)<\/button><span class=\"hover-text\"><span class=\"marginpar\">Wir k\u00f6nnen f\u00fcr jedes <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> einfach <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>L<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><mi>\ud835\udf00<\/mi><\/math> verwenden. (Wobei wir <math display=\"inline\"><mi>L<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/math> im Z\u00e4hler verwenden um auch im eigenartigen Fall <math display=\"inline\"><mi>L<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> eine vern\u00fcnftige Definition f\u00fcr <math display=\"inline\"><mi>\u03b4<\/mi><\/math> zu erhalten.)<\/span><\/span>). Unter gewissen Bedingungen an den metrischen Raum <math display=\"inline\"><mi>X<\/mi><\/math> sind stetige Funktionen gleichm\u00e4ssig stetig (zum Beispiel falls <math display=\"inline\"><mi>X<\/mi><\/math> ein kompaktes Intervall ist); wir werden im zweiten Semester darauf zur\u00fcckkommen. <a id=\"x1-151002r151\"><\/a> <\/p> <h4 id=\"z0ff499c5ef5a\" class=\"subsectionHead\"><span class=\"titlemark\">5.4.3 <\/span> <a id=\"x1-1520003\"><\/a>Konstruktion von stetigen Funktionen<\/h4> <p class=\"noindent\">Wie schon in Abschnitt <a href=\"..\/..\/chapter\/stetigkeit#x1-940005\">3.5<\/a> lassen sich f\u00fcr stetigen Funktionen verschiedene Operationen durchf\u00fchren, die wiederum zu weiteren stetigen Funktionen f\u00fchren. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z58ed13307753\"> <a id=\"x1-152001r53\"><\/a> <span class=\"ecbx-1095\">Proposition 5.53 <\/span>(Stetige Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>Y<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>Z<\/mi><\/math> <span class=\"ecti-1095\">metrische R<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ume.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Y<\/mi> <\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>Y<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Z<\/mi><\/math> <span class=\"ecti-1095\">stetig sind, dann ist auch <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>Z<\/mi><\/math> <span class=\"ecti-1095\">stetig.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Eine Funktion <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">ist genau dann stetig, wenn die Komponenten<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><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> <span class=\"ecti-1095\">stetig sind,<\/span> <span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">:<\/mo> <msup><mrow><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>d<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><mo class=\"MathClass-rel\">\u21a6<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <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> <span class=\"ecti-1095\">die Projektion<\/span> <span class=\"ecti-1095\">auf die <\/span><math display=\"inline\"><mi>j<\/mi><\/math><span class=\"ecti-1095\">-te<\/span> <span class=\"ecti-1095\">Koordinate bezeichnet.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Addition <\/span><math display=\"inline\"><mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">ist stetig.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Multiplikation <\/span><math display=\"inline\"><mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-punc\">:<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">ist stetig.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(v)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Kehrwertfunktion <\/span><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-punc\">:<\/mo> <mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u00d7<\/mo><\/mrow><\/msup><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>z<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">ist stetig.<\/span><\/dd><\/dl> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir verwenden im Beweis mehrmals die Charakterisierung von Stetigkeit mittels Folgen aus Proposition&nbsp;<a href=\"..\/..\/chapter\/stetigkeit#x1-150003r50\">5.50<\/a>.                                                                                                                                                                           <\/p><p class=\"indent\">F\u00fcr (i) sei <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> eine konvergente Folge in <math display=\"inline\"><mi>X<\/mi><\/math> mit Grenzwert <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> Da <math display=\"inline\"><mi>f<\/mi><\/math> stetig ist, konvergiert die Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> (nach Proposition&nbsp;<a href=\"..\/..\/chapter\/stetigkeit#x1-150003r50\">5.50<\/a>) gegen <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Nach Stetigkeit von <math display=\"inline\"><mi>g<\/mi><\/math> konvergiert <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> gegen <span class=\"maperiod\"><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Dies aber impliziert die Aussage in (i) wegen Proposition&nbsp;<a href=\"..\/..\/chapter\/stetigkeit#x1-150003r50\">5.50<\/a>. <\/p><p class=\"indent\">F\u00fcr (ii) bemerken wir zuerst, dass die Projektion <math display=\"inline\"><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">:<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> wegen <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>v<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">\u2225<\/mo><mi>v<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>w<\/mi><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/math> f\u00fcr <math display=\"inline\"><mi>v<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi> <\/mrow> <\/msup> <\/math> Lipschitz-stetig mit Lipschitz-Konstante <math display=\"inline\"><mn>1<\/mn><\/math> ist (bez\u00fcglich der Metrik auf <math display=\"inline\"><msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math> induziert durch <math display=\"inline\"><mo class=\"MathClass-rel\">\u2225<\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><msub><mrow><mo class=\"MathClass-rel\">\u2225<\/mo><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msub><\/math>). Daher folgt aus (i) und Stetigkeit von <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mi>d<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> dass auch <math display=\"inline\"><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>C<\/mi><\/math> stetig ist f\u00fcr jedes <span class=\"maperiod\"><math display=\"inline\"><mi>j<\/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>d<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math><\/span><span class=\"period\">.<\/span> Sei nun umgekehrt <math display=\"inline\"><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><\/math> f\u00fcr jedes <math display=\"inline\"><mi>j<\/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>d<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> stetig. Sei <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <\/math> eine konvergente Folge in <math display=\"inline\"><mi>X<\/mi><\/math> mit Grenzwert <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math><\/span><span class=\"period\">.<\/span> Also konvergiert <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> f\u00fcr alle <math display=\"inline\"><mi>j<\/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>d<\/mi><mo class=\"MathClass-close\">}<\/mo><\/math> nach <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>\u03c0<\/mi><\/mrow><mrow><mi>j<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> was nach Proposition&nbsp;<a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-148002r44\">5.44<\/a> impliziert, dass <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> nach <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> konvergiert. Dies beweist (ii). <\/p><p class=\"indent\">F\u00fcr <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> gilt <\/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><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>w<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>2<\/mn><mo class=\"MathClass-rel\">\u2225<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2225<\/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\">Daher ist die Addition Lipschitz-stetig mit Lipschitz-Konstante <span class=\"maperiod\"><math display=\"inline\"><mn>2<\/mn><\/math><\/span><span class=\"period\">,<\/span> und (iii) folgt. <\/p><p class=\"indent\">F\u00fcr <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mi>w<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mi>w<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mi>w<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/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\">Sei nun <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">\u2225<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>t<\/mi><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2225<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> min<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi>\ud835\udf00<\/mi><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi class=\"qopname\">max<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-punc\">,<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dann gilt <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><\/math> und daher                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><mi>w<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>w<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>w<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Dies zeigt Stetigkeit der Multiplikation in (iv). <\/p><p class=\"indent\">F\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u00d7<\/mo><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u2102<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u00d7<\/mo> <\/mrow> <\/msup> <\/math> mit <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi class=\"qopname\"> min<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>\ud835\udf00<\/mi><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> gilt <\/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>z<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2265<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow> <mo class=\"MathClass-rel\">&gt;<\/mo><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und daher <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><msup><mrow><mi>z<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <msubsup><mrow><mi>z<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>z<\/mi><msub><mrow><mi>z<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>z<\/mi><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>2<\/mn><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/mrow><\/msup><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><mi>\ud835\udf00<\/mi><mo class=\"MathClass-rel\">|<\/mo><msub><mrow><mi>z<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Da <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> beliebig war, zeigt dies die Stetigkeit der Kehrwertabbildung in (v). <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z20d04ce420cf\"> <a id=\"x1-152007r54\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 5.54 <\/span>(Distanzfunktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> d<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein metrischer Raum. In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir den Abstand von Teilmengen von<\/span> <math display=\"inline\"><mi>X<\/mi><\/math> <span class=\"ecti-1095\">zu Punkten<\/span> <span class=\"ecti-1095\">diskutieren. Zu <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>X<\/mi><\/math> <span class=\"ecti-1095\">nicht-leer definieren wir<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">d<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> inf<\/mi><mo>  <\/mo> <\/mrow><mrow><mi>a<\/mi><mo class=\"MathClass-rel\">\u2208<\/mo><mi>A<\/mi><\/mrow><\/munder><mi class=\"qopname\"> d<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Zeigen Sie, dass die Funktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>A<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">Lipschitz-stetig mit Lipschitz-Konstante <\/span><math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">ist.<\/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\">Sie k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/metrische-raeume#x1-140009r14\"><span class=\"ecti-1095\">5.14<\/span><\/a> <span class=\"ecti-1095\">verwenden.<\/span><\/p><\/details>  <\/div> <a id=\"x1-152008r149\"><\/a> \n","protected":false},"author":1089,"menu_order":4,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-65","chapter","type-chapter","status-publish","hentry"],"part":61,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/65","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\/65\/revisions"}],"part":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/parts\/61"}],"metadata":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/65\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=65"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=65"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=65"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=65"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}