{"id":45,"date":"2021-12-15T09:53:04","date_gmt":"2021-12-15T09:53:04","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/reellwertige-funktionen\/"},"modified":"2021-12-15T09:53:04","modified_gmt":"2021-12-15T09:53:04","slug":"reellwertige-funktionen","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/reellwertige-funktionen\/","title":{"raw":"Reellwertige Funktionen","rendered":"Reellwertige Funktionen"},"content":{"raw":"\n<style>.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: 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little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z447327bc1d86\" class=\"sectionHead\"><span class=\"titlemark\">3.4 <\/span> <a id=\"x1-910004\"><\/a>Reellwertige Funktionen<\/h3> <p class=\"noindent\">F\u00fcr eine beliebige, nicht-leere Menge <math display=\"inline\"><mi>D<\/mi><\/math> definieren wir die Menge der <math display=\"inline\"><mi>\u211d<\/mi><\/math><span class=\"ecbx-1095\">-wertigen<\/span> oder <span class=\"ecbx-1095\">reellwertigen <\/span>Funktionen auf <math display=\"inline\"><mi>D<\/mi><\/math> als <a id=\"dx1-91001\"><\/a> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>D<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/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\">Die Menge <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> bildet einen Vektorraum \u00fcber <span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> wobei Addition und skalare Multiplikation punktweise gegeben sind durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03b1<\/mi><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> und alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> Funktionen in <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> lassen sich sogar multiplizieren und zwar durch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> (Die Menge&nbsp;<math display=\"inline\"><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> bildet einen kommutativen Ring mit Eins.) Wir sagen, dass <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> eine <span class=\"ecbx-1095\">Nullstelle<\/span> von <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> ist, falls <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> gilt. Die <span class=\"ecbx-1095\">Nullstellenmenge <\/span>von <math display=\"inline\"><mi>f<\/mi><\/math> ist durch <math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> definiert. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z15310edfc4d1\"> <a id=\"x1-91002r35\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.35 <\/span>(Nullstellenmenge eines Produkts)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">die Menge der Nullstellen von <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">beziehungsweise <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Was ist die Nullstellenmenge von <\/span><span class=\"maendquote\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"endquote\">?<\/span> <\/p> <\/div> <p class=\"indent\">Wir definieren auch eine Relation <math display=\"inline\"> <mo class=\"MathClass-rel\">\u2264<\/mo><\/math> (tats\u00e4chlich eine Ordnung) auf <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> durch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Wir sagen, dass <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecbx-1095\">nicht-negativ<\/span> ist, falls <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> gilt <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z13792ec56163\"> <a id=\"x1-91003r36\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.36 <\/span>(Ordnung auf <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen          Sie,          dass          die          oben          definierte          Relation<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u2264<\/mo><\/math> <span class=\"ecti-1095\">eine Ordnung    ist    und    dass    diese    Ordnung    genau    dann    linear    ist,    wenn<\/span> <math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">aus genau einem Punkt besteht.<\/span> <\/p> <\/div> <a id=\"x1-91004r90\"><\/a> <h4 id=\"z47be17eebbe0\" class=\"subsectionHead\"><span class=\"titlemark\">3.4.1 <\/span> <a id=\"x1-920001\"><\/a>Beschr\u00e4nktheit<\/h4> <p class=\"noindent\">In diesem und im n\u00e4chsten Abschnitt m\u00f6chten wir jeweils einen wichtigen Begriff zu reellwertigen Funktionen einf\u00fchren. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z2aba7c84a1e3\"> <a id=\"x1-92001r37\"><\/a> <span class=\"ecbx-1095\">Definition 3.37 <\/span>(Beschr\u00e4nktheit von Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Sei <math display=\"inline\"><mi>D<\/mi><\/math> eine nicht-leere Menge und sei <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> eine Funktion. Wir sagen, dass die Funktion <math display=\"inline\"><mi>f<\/mi><\/math> <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecbx-1095\">von oben beschr<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nkt <\/span>ist, falls das Bild <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> von oben beschr\u00e4nkt ist, <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecbx-1095\">von unten beschr<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nkt <\/span>ist, falls das Bild <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> von unten beschr\u00e4nkt ist, <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecbx-1095\">beschr<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nkt <\/span>ist, falls <math display=\"inline\"><mi>f<\/mi><\/math> von oben und von unten beschr\u00e4nkt ist.<\/div><\/div> <\/div> <p class=\"indent\">Wir bemerken, dass eine Teilmenge <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> genau dann beschr\u00e4nkt ist, wenn es ein <math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> gibt, so dass <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>M<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/span> Daher ist eine Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> genau dann beschr\u00e4nkt, wenn es ein <math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> gibt, so dass f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>M<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z7d0e9e7775d2\"> <a id=\"x1-92002r38\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.38 <\/span>(Der Unterraum der beschr\u00e4nkten Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">eine nicht-leere Menge und sei <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\u212c<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">die Menge der beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkten Funktionen von <\/span><math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">nach <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\u212c<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein Unterraum von <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">bildet und dass zu <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u212c<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">auch <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u22c5<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u212c<\/mi><\/math> <span class=\"ecti-1095\">liegt.<\/span> <\/p> <\/div> <a id=\"x1-92003r92\"><\/a> <h4 id=\"zee06be765e30\" class=\"subsectionHead\"><span class=\"titlemark\">3.4.2 <\/span> <a id=\"x1-930002\"><\/a>Monotonie<\/h4> <p class=\"noindent\">In Folgendem wollen wir annehmen, dass die Menge <math display=\"inline\"><mi>D<\/mi><\/math> (der Definitionsbereich der Funktionen, die wir betrachten wollen) eine nicht-leere Teilmenge von <math display=\"inline\"><mi>\u211d<\/mi><\/math> ist. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z9e20b0d6ab6b\"> <a id=\"x1-93001r39\"><\/a> <span class=\"ecbx-1095\">Definition 3.39 <\/span>(Monotonieeigenschaften)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Eine Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> ist <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecbx-1095\">monoton wachsend<\/span>, falls <span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecbx-1095\">streng monoton wachsend<\/span>, falls <span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>y<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecbx-1095\">monoton fallend<\/span>, falls <span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecbx-1095\">streng monoton fallend<\/span>, falls <span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>y<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecbx-1095\">monoton<\/span>, falls <math display=\"inline\"><mi>f<\/mi><\/math> monoton wachsend oder monoton fallend ist, <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecbx-1095\">streng monoton<\/span>, falls <math display=\"inline\"><mi>f<\/mi><\/math> streng monoton wachsend oder streng monoton fallend ist.<\/div><\/div> <\/div> <p class=\"indent\">Eine streng monotone Funktion ist per Definition auch monoton; die Bezeichnung \u201estreng\u201c ist also passend. Wir betrachten nun ein paar elementare Beispiele. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z24ced7fe28bd\"> <a id=\"x1-93002r40\"><\/a> <span class=\"ecbx-1095\">Beispiel 3.40.<\/span> <\/h4> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Die Funktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist streng monoton wachsend. Die Funktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2264<\/mo><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist jedoch streng monoton fallend und <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">hat keine Monotonieeigenschaft. Dies zeigt, dass Monotonie vom Definitionsbereich abh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngt.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Die Funktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist streng monoton wachsend.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Allgemeiner gilt: F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">ist die Funktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist genau dann (streng) monoton, wenn <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">ungerade ist.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Die Abrundungsfunktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mo class=\"MathClass-open\">\u230a<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">\u230b<\/mo><mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist monoton wachsend, aber nicht streng monoton wachsend (wieso?).<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Eine konstante Funktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein (festes) <\/span><math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist sowohl monoton fallend als auch monoton wachsend.<\/span><\/div><\/div> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zbfadc5128154\"> <a id=\"x1-93003r41\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.41.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Beweisen Sie die Behauptungen in Beispiel <\/span><a href=\"..\/..\/chapter\/reellwertige-funktionen#x1-93002r40\"><span class=\"ecti-1095\">3.40<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p> <\/div> <p class=\"indent\">Eine streng monotone Funktion ist stets injektiv. Sie braucht jedoch nicht surjektiv zu sein. Beispielsweise ist die Funktion                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mi>x<\/mi> <\/mtd> <mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>0<\/mn><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">streng monoton wachsend, aber nicht surjektiv (<math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> liegt nicht im Bild). <\/p> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=351&amp;url=Pictures\/funktionen_R\/Monotonie\/mono-notsurj.pdf\"><img id=\"z0dfe015c1dac\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/funktionen_R\/Monotonie\/mono-notsurj.svg\" width=\"351\"><\/div>  <\/div> <p class=\"indent\">Verlangt man jedoch von einer streng monotonen Funktion, dass sie \u201enicht springt\u201c, so kann man in vielen F\u00e4llen Surjektivit\u00e4t zeigen. Wir haben bereits ein Beispiel daf\u00fcr gesehen als wir die Existenz der Wurzelfunktion zeigten (siehe \u00dcbung <a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-48001r11\">2.11<\/a>). Den dazu notwendigen Begriff des \u201eNicht-Springens\u201c besprechen wir im n\u00e4chsten Abschnitt. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"za7394dacec4f\"> <a id=\"x1-93004r42\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.42 <\/span>(Alternative Charakterisierung von Monotonie und strenger Monotonie)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Teilmenge mit <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>D<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2265<\/mo> <mn>3<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Funktion. Betrachten Sie die folgenden Aussagen <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">beschreiben Sie ihre Bedeutung in Worten, und geben Sie einen Beweis von drei Ihrer<\/span> <span class=\"ecti-1095\">Behauptungen.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>y<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>z<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2228<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>z<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2228<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(v)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>z<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2228<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(vi)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>z<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2228<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><\/math><\/dd><\/dl> <p class=\"noindent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Teill<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Die Aussagen in (i) und (ii) beschreiben beide, dass<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">streng<\/span> <span class=\"ecti-1095\">monoton wachsend ist. Dies mag bei (ii) etwas <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berraschen und liegt daran, dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine Funktion<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">mit der Eigenschaft<\/span> <span class=\"ecti-1095\">in (ii) und f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">sowohl<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math> <span class=\"ecti-1095\">als<\/span> <span class=\"ecti-1095\">auch <\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi><\/math> <span class=\"ecti-1095\">folgt.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Die Aussage in (iii) beschreibt, dass <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">streng monoton ist. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den Beweis, dass (iii) strenge Monotonie impliziert, w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlen Sie am besten zuerst zwei<\/span> <span class=\"ecti-1095\">feste Elemente <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mi>D<\/mi><\/math><span class=\"ecti-1095\">. Da es einen<\/span> <span class=\"ecti-1095\">dritten Punkt in <\/span><math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">gibt, muss entweder <\/span><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> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">oder <\/span><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> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gelten. Wir<\/span> <span class=\"ecti-1095\">nehmen <\/span><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> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">an und<\/span> <span class=\"ecti-1095\">werden zeigen, dass <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">streng monoton wachsend ist. (Der Beweis im zweiten Fall ergibt sich hieraus durch Betrachten von<\/span> <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><\/math><span class=\"ecti-1095\">.) F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r beliebige<\/span> <span class=\"ecti-1095\">Element <\/span><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">folgt daraus:<\/span> <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mspace class=\"thickpace\" width=\"0.28em\" \/> <mo class=\"MathClass-rel\">\u21d2<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/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> <span class=\"ecti-1095\">wenn wir <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">setzen und die Vorraussetzung in (iii) anwenden (wobei die zweite M<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">glichkeit durch<\/span> <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> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ausgeschlossen ist).<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>a<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <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-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">denn f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist dies unsere Annahme, ansonsten k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">setzen falls <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">aber <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><\/math> <span class=\"ecti-1095\">falls <\/span><math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>a<\/mi><\/math> <span class=\"ecti-1095\">und wiederum (iii) anwenden.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">denn f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">oder <\/span><math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist dies auf Grund der ersten beiden Punkte bereits bekannt und ansonsten k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>z<\/mi><\/math> <span class=\"ecti-1095\">eindeutig so definieren, dass <\/span><math display=\"inline\"><mo class=\"MathClass-open\">{<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">}<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">und durch Anwendung von (iii) ergibt sich die gew<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nschte Aussage.<\/span><\/div><\/div> <p class=\"indent\"><span class=\"ecti-1095\">Die Aussage in (iv) und auch in (vi) beschreibt Monotonie von<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Die Aussage in (v) ist hingegen unsinnig und trifft auf keine Funktion zu, denn aus<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>z<\/mi><\/math> <span class=\"ecti-1095\">muss<\/span> <span class=\"ecti-1095\">nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rlich <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">folgen. <\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"ze4fb78323bf9\"> <a id=\"x1-93011r43\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.43 <\/span>(Monotonie unter Summen und Produkten)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Teilmenge und seien <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">streng monoton wachsend. Zeigen Sie, dass <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">streng monoton wachsend ist und dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">die Funktion <\/span><math display=\"inline\"><mi>a<\/mi><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">streng monoton wachsend ist falls <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">und streng monoton fallend ist falls <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">streng monoton wachsend ist, falls <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z4ca16d0fc3a3\"> <a id=\"x1-93012r44\"><\/a> <span class=\"ecbx-1095\">Applet 3.44 <\/span>(Monotonie von Einschr\u00e4nkungen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"530px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/FcDfJMp5\/width\/688\/height\/530\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Bei vielen aber nicht allen Funktion <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">(definiert auf Teilmengen von <\/span><math display=\"inline\"><mi>\u211d<\/mi><\/math><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">erh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lt man eine monotone Funktion mittels Einschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkung von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">auf kleinere Intervalle.<\/span> <\/p> <\/div> <a id=\"x1-93013r91\"><\/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: 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}\n\n.entry-title{display: none}\n\ndiv.newtheorem { margin-bottom: 2em; margin-top: 2em; border: 1px solid #333; background: #c7e4da; border-color: #4eb79e;}\ndiv.newtheorem h3 { background: #4eb79e; color: white; padding: 0px 15px 0px 15px; margin-top: 12px}\ndiv.newtheorem p { padding: 15px 15px 15px 15px; }\n\ndiv.newtheorem p span.head .ecbx-1095{font-weight: bold}\ndiv.newtheorem p .ecti-1095{font-style: italic}\ndiv.newtheorem div.custom-itemize{font-style: italic}\ndiv.quote{font-style: italic}\ndiv.newtheorem dl, dl.enumerate {display: grid; grid-template-columns: 5% auto; align-items: start; margin-top: 1em}\ndiv.newtheorem dl dd, dl.enumerate dd {margin-bottom: 0.5em}\ndiv.newtheorem dl dt, dl.enumerate dt {font-weight: normal; margin-top: 0px; text-align: right; margin-right: 15%}\ndiv.newtheorem dl dd {font-style: italic}\ndiv.newtheorem dl dt {font-style: italic}\ndiv.proof p span.ecti-1095 {font-style: italic}\ndiv.figure p img { margin-left: auto; margin-right: auto; display: block; }\ndiv.mefigcentered, div.figure { text-align: center }\n\ndl:after {content:\"\";display:table;clear:both;}\ndd {padding:.5em 0;}\ndl {width:100%;}\ndt, dd {display:inline-block; width:125%;}\ndt {text-align:right; font-weight:bold; clear:left; float:left;}\ndd {width:100%; padding-left:1em; padding-top: 0px; clear:right;}\ndd + dd {float:right; clear:both;}\ndd + dt {clear:both;}\ndt + dt {width: 100%; float: none; padding: 0 70% 0 0;}\ndt + dt + dd {margin-top: -2em;}\ndt + dt + dd + dt {margin-top: 2em;}\n<\/style>\n<style scoped=\"scoped\">\n\/* CSS Analysis-Skript D-Math ETHZ *\/\n\n\/* Uniform Font, also for headers *\/\nh3 {\n\tfont-family: \"Times New Roman\", serif;\n\tmargin-bottom: 35px;\n}\nh4 {\n\tfont-family: \"Times New Roman\", serif;\n}\nh5 {\n\tfont-family: \"Times New Roman\", serif;\n}\n\n\/* Bold font, e.g. for definitions *\/\n.ecbx-1095 {font-weight: 550 ;}\n\n\n\/* Uniform spacing, indent: larger, noindent, enumerate, itemize *\/\np.indent {\n\tmargin: 25px 0px 0px 0px;\n\ttext-indent: 0px; \n}\np.noindent {\n\tmargin: 15px 0px 0px 0px;\n\ttext-indent: 0px; \n}\ndl.enumerate {\n\tmargin: 0px 0px 0px 0px;\n}\ndl.enumerate dt, dl.enumerate dd {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\ndiv.custom-itemize {\n\tmargin: 0px 0px 0px 0px;\n}\ndiv.custom-itemize div.item-head {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n\ttext-align: center;\n}\ndiv.custom-itemize div.item-head:first-of-type {\n\tmargin-top: 0px;\n} \ndiv.custom-itemize div.item-content {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\n.MJXc-display {\n\tmargin: 15px 0px 0px 0px;\n}\n\n\n\n\/* green metheorem\/melemma CSS class for more\/medium important latex-theorem-environments *\/\n\/* metheorem box+header *\/\ndiv.metheorem {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #c7e4da;\n}\ndiv.metheorem h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* melemma box+header *\/\ndiv.melemma {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #F2F2F2;\n}\ndiv.melemma h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* meexample box+header *\/\ndiv.meexample {\n    margin-bottom: 30px;\n    margin-top: 30px;\n\tpadding: 0px 15px 15px 15px;\n\tborder-color: gainsboro;\n\tborder-style: solid;\n\tborder-width: thin;\n}\ndiv.meexample h4 {\n\tfont-size: inherit;\n\tfont-weight: bold;\n    padding: 15px 0px 0px 0px;\n\tmargin-top: 0px;\n\tmargin-bottom: 5px;\n}\ndiv.meexample h4+p.noindent, div.meexample h4+p.indent {\n\tmargin-top: 5px;\n\ttext-indent: 0px;\n}\n\/* padding and margins for stuff inside these boxes, CSS-selector &gt; doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z447327bc1d86\" class=\"sectionHead\"><span class=\"titlemark\">3.4 <\/span> <a id=\"x1-910004\"><\/a>Reellwertige Funktionen<\/h3> <p class=\"noindent\">F\u00fcr eine beliebige, nicht-leere Menge <math display=\"inline\"><mi>D<\/mi><\/math> definieren wir die Menge der <math display=\"inline\"><mi>\u211d<\/mi><\/math><span class=\"ecbx-1095\">-wertigen<\/span> oder <span class=\"ecbx-1095\">reellwertigen <\/span>Funktionen auf <math display=\"inline\"><mi>D<\/mi><\/math> als <a id=\"dx1-91001\"><\/a> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mi>D<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/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\">Die Menge <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> bildet einen Vektorraum \u00fcber <span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> wobei Addition und skalare Multiplikation punktweise gegeben sind durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03b1<\/mi><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> und alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> Funktionen in <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> lassen sich sogar multiplizieren und zwar durch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> (Die Menge&nbsp;<math display=\"inline\"><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> bildet einen kommutativen Ring mit Eins.) Wir sagen, dass <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> eine <span class=\"ecbx-1095\">Nullstelle<\/span> von <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> ist, falls <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> gilt. Die <span class=\"ecbx-1095\">Nullstellenmenge <\/span>von <math display=\"inline\"><mi>f<\/mi><\/math> ist durch <math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> definiert. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z15310edfc4d1\"> <a id=\"x1-91002r35\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.35 <\/span>(Nullstellenmenge eines Produkts)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>N<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">die Menge der Nullstellen von <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">beziehungsweise <\/span><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Was ist die Nullstellenmenge von <\/span><span class=\"maendquote\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"endquote\">?<\/span> <\/p> <\/div> <p class=\"indent\">Wir definieren auch eine Relation <math display=\"inline\"> <mo class=\"MathClass-rel\">\u2264<\/mo><\/math> (tats\u00e4chlich eine Ordnung) auf <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> durch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Wir sagen, dass <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecbx-1095\">nicht-negativ<\/span> ist, falls <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> gilt <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z13792ec56163\"> <a id=\"x1-91003r36\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.36 <\/span>(Ordnung auf <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen          Sie,          dass          die          oben          definierte          Relation<\/span> <math display=\"inline\"><mo class=\"MathClass-rel\">\u2264<\/mo><\/math> <span class=\"ecti-1095\">eine Ordnung    ist    und    dass    diese    Ordnung    genau    dann    linear    ist,    wenn<\/span> <math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">aus genau einem Punkt besteht.<\/span> <\/p> <\/div> <a id=\"x1-91004r90\"><\/a> <h4 id=\"z47be17eebbe0\" class=\"subsectionHead\"><span class=\"titlemark\">3.4.1 <\/span> <a id=\"x1-920001\"><\/a>Beschr\u00e4nktheit<\/h4> <p class=\"noindent\">In diesem und im n\u00e4chsten Abschnitt m\u00f6chten wir jeweils einen wichtigen Begriff zu reellwertigen Funktionen einf\u00fchren. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z2aba7c84a1e3\"> <a id=\"x1-92001r37\"><\/a> <span class=\"ecbx-1095\">Definition 3.37 <\/span>(Beschr\u00e4nktheit von Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Sei <math display=\"inline\"><mi>D<\/mi><\/math> eine nicht-leere Menge und sei <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> eine Funktion. Wir sagen, dass die Funktion <math display=\"inline\"><mi>f<\/mi><\/math> <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecbx-1095\">von oben beschr<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nkt <\/span>ist, falls das Bild <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> von oben beschr\u00e4nkt ist, <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecbx-1095\">von unten beschr<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nkt <\/span>ist, falls das Bild <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> von unten beschr\u00e4nkt ist, <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecbx-1095\">beschr<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">nkt <\/span>ist, falls <math display=\"inline\"><mi>f<\/mi><\/math> von oben und von unten beschr\u00e4nkt ist.<\/div><\/div> <\/div> <p class=\"indent\">Wir bemerken, dass eine Teilmenge <math display=\"inline\"><mi>A<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> genau dann beschr\u00e4nkt ist, wenn es ein <math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> gibt, so dass <math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2264<\/mo> <mi>M<\/mi><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>A<\/mi><\/math><\/span><span class=\"period\">.<\/span> Daher ist eine Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> genau dann beschr\u00e4nkt, wenn es ein <math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> gibt, so dass f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>M<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z7d0e9e7775d2\"> <a id=\"x1-92002r38\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.38 <\/span>(Der Unterraum der beschr\u00e4nkten Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">eine nicht-leere Menge und sei <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\u212c<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">die Menge der beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkten Funktionen von <\/span><math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">nach <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\u212c<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein Unterraum von <\/span><math display=\"inline\"><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">bildet und dass zu <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u212c<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">auch <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u22c5<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u212c<\/mi><\/math> <span class=\"ecti-1095\">liegt.<\/span> <\/p> <\/div> <a id=\"x1-92003r92\"><\/a> <h4 id=\"zee06be765e30\" class=\"subsectionHead\"><span class=\"titlemark\">3.4.2 <\/span> <a id=\"x1-930002\"><\/a>Monotonie<\/h4> <p class=\"noindent\">In Folgendem wollen wir annehmen, dass die Menge <math display=\"inline\"><mi>D<\/mi><\/math> (der Definitionsbereich der Funktionen, die wir betrachten wollen) eine nicht-leere Teilmenge von <math display=\"inline\"><mi>\u211d<\/mi><\/math> ist. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z9e20b0d6ab6b\"> <a id=\"x1-93001r39\"><\/a> <span class=\"ecbx-1095\">Definition 3.39 <\/span>(Monotonieeigenschaften)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Eine Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> ist <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecbx-1095\">monoton wachsend<\/span>, falls <span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecbx-1095\">streng monoton wachsend<\/span>, falls <span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>y<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecbx-1095\">monoton fallend<\/span>, falls <span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecbx-1095\">streng monoton fallend<\/span>, falls <span class=\"maperiod\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>y<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecbx-1095\">monoton<\/span>, falls <math display=\"inline\"><mi>f<\/mi><\/math> monoton wachsend oder monoton fallend ist, <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecbx-1095\">streng monoton<\/span>, falls <math display=\"inline\"><mi>f<\/mi><\/math> streng monoton wachsend oder streng monoton fallend ist.<\/div><\/div> <\/div> <p class=\"indent\">Eine streng monotone Funktion ist per Definition auch monoton; die Bezeichnung \u201estreng\u201c ist also passend. Wir betrachten nun ein paar elementare Beispiele. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z24ced7fe28bd\"> <a id=\"x1-93002r40\"><\/a> <span class=\"ecbx-1095\">Beispiel 3.40.<\/span> <\/h4> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Die Funktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist streng monoton wachsend. Die Funktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">\u2264<\/mo><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist jedoch streng monoton fallend und <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">hat keine Monotonieeigenschaft. Dies zeigt, dass Monotonie vom Definitionsbereich abh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngt.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Die Funktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist streng monoton wachsend.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Allgemeiner gilt: F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">ist die Funktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist genau dann (streng) monoton, wenn <\/span><math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">ungerade ist.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Die Abrundungsfunktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mo class=\"MathClass-open\">\u230a<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">\u230b<\/mo><mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist monoton wachsend, aber nicht streng monoton wachsend (wieso?).<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Eine konstante Funktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein (festes) <\/span><math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist sowohl monoton fallend als auch monoton wachsend.<\/span><\/div><\/div> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zbfadc5128154\"> <a id=\"x1-93003r41\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.41.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Beweisen Sie die Behauptungen in Beispiel <\/span><a href=\"..\/..\/chapter\/reellwertige-funktionen#x1-93002r40\"><span class=\"ecti-1095\">3.40<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p> <\/div> <p class=\"indent\">Eine streng monotone Funktion ist stets injektiv. Sie braucht jedoch nicht surjektiv zu sein. Beispielsweise ist die Funktion                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mi>x<\/mi> <\/mtd> <mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>0<\/mn><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">streng monoton wachsend, aber nicht surjektiv (<math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac><\/math> liegt nicht im Bild). <\/p> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=351&amp;url=Pictures\/funktionen_R\/Monotonie\/mono-notsurj.pdf\"><img decoding=\"async\" id=\"z0dfe015c1dac\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/funktionen_R\/Monotonie\/mono-notsurj.svg\" width=\"351\" \/><\/div>  <\/div> <p class=\"indent\">Verlangt man jedoch von einer streng monotonen Funktion, dass sie \u201enicht springt\u201c, so kann man in vielen F\u00e4llen Surjektivit\u00e4t zeigen. Wir haben bereits ein Beispiel daf\u00fcr gesehen als wir die Existenz der Wurzelfunktion zeigten (siehe \u00dcbung <a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-48001r11\">2.11<\/a>). Den dazu notwendigen Begriff des \u201eNicht-Springens\u201c besprechen wir im n\u00e4chsten Abschnitt. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"za7394dacec4f\"> <a id=\"x1-93004r42\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.42 <\/span>(Alternative Charakterisierung von Monotonie und strenger Monotonie)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Teilmenge mit <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>D<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2265<\/mo> <mn>3<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Funktion. Betrachten Sie die folgenden Aussagen <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">beschreiben Sie ihre Bedeutung in Worten, und geben Sie einen Beweis von drei Ihrer<\/span> <span class=\"ecti-1095\">Behauptungen.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>y<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d4<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>z<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2228<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>z<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2228<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(v)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>z<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2228<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(vi)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi class=\"MathClass-op\">\u2200<\/mi><mo> <\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>z<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2228<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><\/math><\/dd><\/dl> <div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Teill<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">sung.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Die Aussagen in (i) und (ii) beschreiben beide, dass<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">streng<\/span> <span class=\"ecti-1095\">monoton wachsend ist. Dies mag bei (ii) etwas <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berraschen und liegt daran, dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine Funktion<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">mit der Eigenschaft<\/span> <span class=\"ecti-1095\">in (ii) und f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">sowohl<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>y<\/mi><\/math> <span class=\"ecti-1095\">als<\/span> <span class=\"ecti-1095\">auch <\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>x<\/mi><\/math> <span class=\"ecti-1095\">folgt.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Die Aussage in (iii) beschreibt, dass <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">streng monoton ist. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den Beweis, dass (iii) strenge Monotonie impliziert, w<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlen Sie am besten zuerst zwei<\/span> <span class=\"ecti-1095\">feste Elemente <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mi>D<\/mi><\/math><span class=\"ecti-1095\">. Da es einen<\/span> <span class=\"ecti-1095\">dritten Punkt in <\/span><math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">gibt, muss entweder <\/span><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> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">oder <\/span><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> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gelten. Wir<\/span> <span class=\"ecti-1095\">nehmen <\/span><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> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">an und<\/span> <span class=\"ecti-1095\">werden zeigen, dass <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">streng monoton wachsend ist. (Der Beweis im zweiten Fall ergibt sich hieraus durch Betrachten von<\/span> <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><\/math><span class=\"ecti-1095\">.) F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r beliebige<\/span> <span class=\"ecti-1095\">Element <\/span><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">folgt daraus:<\/span> <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mspace class=\"thickpace\" width=\"0.28em\" \/> <mo class=\"MathClass-rel\">\u21d2<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/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> <span class=\"ecti-1095\">wenn wir <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">setzen und die Vorraussetzung in (iii) anwenden (wobei die zweite M<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">glichkeit durch<\/span> <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> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ausgeschlossen ist).<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>a<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <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-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">denn f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist dies unsere Annahme, ansonsten k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">setzen falls <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">aber <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>a<\/mi><\/math> <span class=\"ecti-1095\">falls <\/span><math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>a<\/mi><\/math> <span class=\"ecti-1095\">und wiederum (iii) anwenden.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo> <mspace class=\"thickpace\" width=\"0.28em\" \/> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">denn f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">oder <\/span><math display=\"inline\"><mi>b<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist dies auf Grund der ersten beiden Punkte bereits bekannt und ansonsten k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>z<\/mi><\/math> <span class=\"ecti-1095\">eindeutig so definieren, dass <\/span><math display=\"inline\"><mo class=\"MathClass-open\">{<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">}<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">{<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">}<\/mo><\/math> <span class=\"ecti-1095\">und durch Anwendung von (iii) ergibt sich die gew<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">nschte Aussage.<\/span><\/div><\/div> <p class=\"indent\"><span class=\"ecti-1095\">Die Aussage in (iv) und auch in (vi) beschreibt Monotonie von<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Die Aussage in (v) ist hingegen unsinnig und trifft auf keine Funktion zu, denn aus<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>z<\/mi><\/math> <span class=\"ecti-1095\">muss<\/span> <span class=\"ecti-1095\">nat<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rlich <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">folgen. <\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"ze4fb78323bf9\"> <a id=\"x1-93011r43\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.43 <\/span>(Monotonie unter Summen und Produkten)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Teilmenge und seien <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">streng monoton wachsend. Zeigen Sie, dass <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">streng monoton wachsend ist und dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">die Funktion <\/span><math display=\"inline\"><mi>a<\/mi><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">streng monoton wachsend ist falls <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">und streng monoton fallend ist falls <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">streng monoton wachsend ist, falls <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z4ca16d0fc3a3\"> <a id=\"x1-93012r44\"><\/a> <span class=\"ecbx-1095\">Applet 3.44 <\/span>(Monotonie von Einschr\u00e4nkungen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <div class=\"wp-nocaption \"><\/div><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"530px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/FcDfJMp5\/width\/688\/height\/530\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Bei vielen aber nicht allen Funktion <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">(definiert auf Teilmengen von <\/span><math display=\"inline\"><mi>\u211d<\/mi><\/math><span class=\"ecti-1095\">)<\/span> <span class=\"ecti-1095\">erh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lt man eine monotone Funktion mittels Einschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkung von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">auf kleinere Intervalle.<\/span> <\/p> <\/div> <a id=\"x1-93013r91\"><\/a> 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