{"id":47,"date":"2021-12-15T09:53:05","date_gmt":"2021-12-15T09:53:05","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/der-zwischenwertsatz\/"},"modified":"2021-12-15T09:53:05","modified_gmt":"2021-12-15T09:53:05","slug":"der-zwischenwertsatz","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/der-zwischenwertsatz\/","title":{"raw":"Der Zwischenwertsatz","rendered":"Der Zwischenwertsatz"},"content":{"raw":"\n<style>.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; border-left: 0; border-right: 0; }\ncenter { margin-top:1em; margin-bottom:1em; }\ntd center { margin-top:0em; margin-bottom:0em; }\n.Canvas { position:relative; }\nmath { text-indent: 0em; }\nli p.indent { text-indent: 0em }\nli p:first-child{ margin-top:0em; }\nli p:last-child, li div:last-child { margin-bottom:0.5em; }\nli p~ul:last-child, li p~ol:last-child{ margin-bottom:0.5em; }\n.enumerate1 {list-style-type:decimal;}\n.enumerate2 {list-style-type:lower-alpha;}\n.enumerate3 {list-style-type:lower-roman;}\n.enumerate4 {list-style-type:upper-alpha;}\n.obeylines-h,.obeylines-v {white-space: nowrap; }\ndiv.obeylines-v p { margin-top:0; margin-bottom:0; }\n.overline{ text-decoration:overline; }\n.overline img{ border-top: 1px solid black; }\ntd.displaylines {text-align:center; white-space:nowrap;}\n.centerline {text-align:center;}\n.rightline {text-align:right;}\npre.verbatim {font-family: monospace,monospace; text-align:left; clear:both; }\n.fbox {padding-left:3.0pt; padding-right:3.0pt; text-indent:0pt; border:solid black 0.4pt; }\ndiv.fbox {display:table}\ndiv.center div.fbox {text-align:center; clear:both; padding-left:3.0pt; padding-right:3.0pt; text-indent:0pt; border:solid black 0.4pt; }\ndiv.minipage{width:100%;}\ndiv.center, div.center div.center {text-align: center; margin-left:1em; margin-right:1em;}\ndiv.center {text-align: left;}\ndiv.flushright, div.flushright div.flushright {text-align: right;}\ndiv.flushright div {text-align: left;}\ndiv.flushleft {text-align: left;}\n.underline{ text-decoration:underline; }\n.underline img{ border-bottom: 1px solid black; margin-bottom:1pt; }\n.framebox-c, .framebox-l, .framebox-r { padding-left:3.0pt; padding-right:3.0pt; text-indent:0pt; border:solid black 0.4pt; }\n.framebox-c {text-align:center;}\n.framebox-l {text-align:left;}\n.framebox-r {text-align:right;}\nspan.thank-mark{ vertical-align: super }\nspan.footnote-mark sup.textsuperscript, span.footnote-mark a sup.textsuperscript{ font-size:80%; }\ndiv.tabular, div.center div.tabular {text-align: center; margin-top:0.5em; margin-bottom:0.5em; }\ntable.tabular td p{margin-top:0em;}\ntable.tabular {margin-left: auto; margin-right: auto;}\ntd p:first-child{ margin-top:0em; }\ntd p:last-child{ margin-bottom:0em; }\ndiv.td00{ margin-left:0pt; margin-right:0pt; }\ndiv.td01{ margin-left:0pt; margin-right:5pt; }\ndiv.td10{ margin-left:5pt; margin-right:0pt; }\ndiv.td11{ margin-left:5pt; margin-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\ntd.td00{ padding-left:0pt; padding-right:0pt; }\ntd.td01{ padding-left:0pt; padding-right:5pt; }\ntd.td10{ padding-left:5pt; padding-right:0pt; }\ntd.td11{ padding-left:5pt; padding-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\n.hline hr, .cline hr{ height : 0px; margin:0px; }\n.hline td, .cline td{ padding: 0; }\n.hline hr, .cline hr{border:none;border-top:1px solid black;}\n.tabbing-right {text-align:right;}\ndiv.float, div.figure {margin-left: auto; margin-right: auto;}\ndiv.float img {text-align:center;}\ndiv.figure img {text-align:center;}\n.marginpar,.reversemarginpar {width:20%; float:right; text-align:left; margin-left:auto; margin-top:0.5em; font-size:85%; text-decoration:underline;}\n.marginpar p,.reversemarginpar p{margin-top:0.4em; margin-bottom:0.4em;}\n.reversemarginpar{float:left;}\n.equation td{text-align:center; vertical-align:middle; }\ntd.eq-no{ width:5%; }\ntable.equation { width:100%; }\ndiv.math-display, div.par-math-display{text-align:center;}\nmtr.hline mtd{ border-bottom:black solid 1px; padding-top:2px; padding-bottom:0em; }\nmtr.hline mtd mo{ display:none }\nmath .texttt { font-family: monospace; }\nmath .textit { font-style: italic; }\nmath .textsl { font-style: oblique; }\nmath .textsf { font-family: sans-serif; }\nmath .textbf { font-weight: bold; }\nmo.MathClass-op + mi{margin-left:0.3em}\nmi + mo.MathClass-op{margin-left:0.3em}\n math mstyle[mathvariant=\"bold\"] { font-weight: bold; font-style: normal; }\n math mstyle[mathvariant=\"normal\"] { font-weight: normal; font-style: normal; }\n.partToc a, .partToc, .likepartToc a, .likepartToc {line-height: 200%; font-weight:bold; font-size:110%;}\n.index-item, .index-subitem, .index-subsubitem {display:block}\ndiv.caption {text-indent:-2em; margin-left:3em; margin-right:1em; text-align:left;}\ndiv.caption span.id{font-weight: bold; white-space: nowrap; }\nh1.partHead{text-align: center}\np.bibitem { text-indent: -2em; margin-left: 2em; margin-top:0.6em; margin-bottom:0.6em; }\np.bibitem-p { text-indent: 0em; margin-left: 2em; margin-top:0.6em; margin-bottom:0.6em; }\n.paragraphHead, .likeparagraphHead { margin-top:2em; font-weight: bold;}\n.subparagraphHead, .likesubparagraphHead { font-weight: bold;}\n.quote {margin-bottom:0.25em; margin-top:0.25em; margin-left:1em; margin-right:1em; text-align:justify;}\n.verse{white-space:nowrap; margin-left:2em}\ndiv.maketitle {text-align:center;}\nh2.titleHead{text-align:center;}\ndiv.maketitle{ margin-bottom: 2em; }\ndiv.author, div.date {text-align:center;}\ndiv.thanks{text-align:left; margin-left:10%; font-size:85%; font-style:italic; }\ndiv.author{white-space: nowrap;}\n.quotation {margin-bottom:0.25em; margin-top:0.25em; margin-left:1em; }\n.abstract p {margin-left:5%; margin-right:5%;}\ndiv.abstract {width:100%;}\ndiv.tabular, div.center div.tabular {text-align: center; margin-top:0.5em; margin-bottom:0.5em; }\ntable.tabular td p{margin-top:0em;}\ntable.tabular {margin-left: auto; margin-right: auto;}\ntd p:first-child{ margin-top:0em; }\ntd p:last-child{ margin-bottom:0em; }\ndiv.td00{ margin-left:0pt; margin-right:0pt; }\ndiv.td01{ margin-left:0pt; margin-right:5pt; }\ndiv.td10{ margin-left:5pt; margin-right:0pt; }\ndiv.td11{ margin-left:5pt; margin-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\ntd.td00{ padding-left:0pt; padding-right:0pt; }\ntd.td01{ padding-left:0pt; padding-right:5pt; }\ntd.td10{ padding-left:5pt; padding-right:0pt; }\ntd.td11{ padding-left:5pt; padding-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\n.hline hr, .cline hr{ height : 0px; margin:0px; }\n.hline td, .cline td{ padding: 0; }\n.hline hr, .cline hr{border:none;border-top:1px solid black;}\n.equation-star td{text-align:center; vertical-align:middle; }\ntable.equation-star { width:100%; border-bottom-color: rgb(255,255,255); }\n#content table.equation-star, #content table.equation-star tbody tr td { border: 0px none rgb(255,255,255); }\nmtd.align-odd{margin-left:2em; text-align:right;}\nmtd.align-even{margin-right:2em; text-align:left;}\n.boxed{border: 1px solid black; padding-left:2px; padding-right:2px;}\n.rotatebox{display: inline-block;}\n.item-head{float:left;width:2em;clear:left;}\n.item-content{margin-left:2em;}\n .foreignobject {line-height:100%; font-size:120%; font-family:STIXgeneral,Times,Symbol,cmr10,CMSY10,CMEX10;padding:0; margin:0; text-align:center; }\nmath {vertical-align:baseline; line-height:100%; font-size:100%; font-family:STIXGeneral,Times,Symbol, cmr10,cmsy10,cmex10,cmmi10; font-style: normal; margin:0; padding:0; }\n\n.entry-title{display: none}\n\ndiv.newtheorem { margin-bottom: 2em; margin-top: 2em; border: 1px solid #333; background: #c7e4da; border-color: #4eb79e;}\ndiv.newtheorem h3 { background: #4eb79e; color: white; padding: 0px 15px 0px 15px; margin-top: 12px}\ndiv.newtheorem p { padding: 15px 15px 15px 15px; }\n\ndiv.newtheorem p span.head .ecbx-1095{font-weight: bold}\ndiv.newtheorem p .ecti-1095{font-style: italic}\ndiv.newtheorem div.custom-itemize{font-style: italic}\ndiv.quote{font-style: italic}\ndiv.newtheorem dl, dl.enumerate {display: grid; grid-template-columns: 5% auto; align-items: start; margin-top: 1em}\ndiv.newtheorem dl dd, dl.enumerate dd {margin-bottom: 0.5em}\ndiv.newtheorem dl dt, dl.enumerate dt {font-weight: normal; margin-top: 0px; text-align: right; margin-right: 15%}\ndiv.newtheorem dl dd {font-style: italic}\ndiv.newtheorem dl dt {font-style: italic}\ndiv.proof p span.ecti-1095 {font-style: italic}\ndiv.figure p img { margin-left: auto; margin-right: auto; display: block; }\ndiv.mefigcentered, div.figure { text-align: center }\n\ndl:after {content:\"\";display:table;clear:both;}\ndd {padding:.5em 0;}\ndl {width:100%;}\ndt, dd {display:inline-block; width:125%;}\ndt {text-align:right; font-weight:bold; clear:left; float:left;}\ndd {width:100%; padding-left:1em; padding-top: 0px; clear:right;}\ndd + dd {float:right; clear:both;}\ndd + dt {clear:both;}\ndt + dt {width: 100%; float: none; padding: 0 70% 0 0;}\ndt + dt + dd {margin-top: -2em;}\ndt + dt + dd + dt {margin-top: 2em;}\n<\/style>\n<style>\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=\"z8d303c0fcb9f\" class=\"sectionHead\"><span class=\"titlemark\">3.6 <\/span> <a id=\"x1-960006\"><\/a>Der Zwischenwertsatz<\/h3> <p class=\"noindent\">In diesem Abschnitt wollen wir einen fundamentalen Satz beweisen, der die Heuristik, dass der Graph einer stetigen Funktion auf einem Intervall \u201eeine durchgehende Kurve\u201c darstellt, formalisiert. Wir sagen, dass eine reelle Zahl <math display=\"inline\"><mi>c<\/mi><\/math> <span class=\"ecbx-1095\">zwischen zwei<\/span> <span class=\"ecbx-1095\">reellen Zahlen <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecbx-1095\">liegt<\/span>, falls <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>c<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> oder <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>c<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> gilt. Wir sagen&nbsp;<math display=\"inline\"><mi>c<\/mi><\/math> liegt <span class=\"ecbx-1095\">echt<\/span> <span class=\"ecbx-1095\">zwischen<\/span><span class=\"ecbx-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecbx-1095\">und<\/span><span class=\"ecbx-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> falls&nbsp;<math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>c<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> oder <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>c<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> ist. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zd703df118694\"> <a id=\"x1-96001r58\"><\/a> <span class=\"ecbx-1095\">Satz 3.58 <\/span>(Zwischenwertsatz)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>I<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein Intervall, <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>I<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine stetige Funktion und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>I<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">zwischen <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gibt es ein <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">zwischen <\/span><math display=\"inline\"><mi>a<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>b<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass <\/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>c<\/mi><\/math> <span class=\"ecti-1095\">gilt.<\/span> <\/p> <\/div> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=575&amp;url=Pictures\/funktionen_R\/zwsatz\/zwsatz.pdf\"><img id=\"z89e665fb5463\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/funktionen_R\/zwsatz\/zwsatz.svg\" width=\"575\"><\/div> <a id=\"x1-96002r6\"><\/a> <a id=\"x1-96003\"><\/a> <br><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;3.6:     <\/span><span class=\"content\">Der     Graph     einer     stetigen     Funktion     kann               keine Spr\u00fcnge machen und die Funktion nimmt alle Werte zwischen               <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math>               und               <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math>               an.                                                                                          &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <p class=\"indent\">Wie wir sehen werden, verwendet der Beweis die Existenz des Supremums (und damit das Vollst\u00e4ndigkeitsaxiom). <\/p><p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir nehmen ohne Beschr\u00e4nkung der Allgemeinheit an, dass <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> und <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> gilt (falls <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> ist, betrachtet man zuerst <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><\/math> und bemerkt, dass die Aussage des Satzes f\u00fcr <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><\/math> die Aussage des Satzes f\u00fcr <math display=\"inline\"><mi>f<\/mi><\/math> impliziert). <\/p><p class=\"indent\">Sei nun <span class=\"maperiod\"><math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> Falls <math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> oder <math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> gilt, sind wir fertig. Also angenommen <span class=\"maperiod\"><math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Wir definieren <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><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\">\u2264<\/mo> <mi>c<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und bemerken, dass <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">,<\/span>                                                                                                                                                                           wodurch <math display=\"inline\"><mi>X<\/mi><\/math> nicht-leer und von oben beschr\u00e4nkt ist. Nach Satz <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\">2.59<\/a> existiert daher <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> Wir werden nun die Stetigkeit von <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> verwenden, um zu zeigen, dass <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>c<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">F\u00fcr jedes <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> gibt es ein <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> so dass f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> gilt <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-96004r6\" \/><mstyle class=\"maketag\"><mtext>(3.6)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Angenommen <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>c<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann folgt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> wegen <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>c<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> Wir wenden nun die Stetigkeit von <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> an und finden f\u00fcr <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>c<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ein <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> wie in (<a href=\"..\/..\/chapter\/der-zwischenwertsatz#x1-96004r6\">3.6<\/a>). Da <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> ist, existiert ein <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> F\u00fcr dieses <math display=\"inline\"><mi>x<\/mi><\/math> gilt dann                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><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><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><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> <mo class=\"MathClass-bin\">+<\/mo> <mi>c<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>c<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Also muss <math display=\"inline\"><mi>x<\/mi><\/math> in <math display=\"inline\"><mi>X<\/mi><\/math> liegen, was aber <math display=\"inline\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>x<\/mi><\/math> widerspricht. <\/p><p class=\"indent\">Angenommen <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>c<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann folgt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>a<\/mi><\/math> wegen <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>c<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wir verwenden wieder die Stetigkeit von <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> und finden zu <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>c<\/mi><\/math> ein <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> mit der Eigenschaft in (<a href=\"..\/..\/chapter\/der-zwischenwertsatz#x1-96004r6\">3.6<\/a>) . F\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> gilt dadurch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><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><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>c<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wodurch <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u2209<\/mo> <mi>X<\/mi><\/math> und daher <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2205<\/mi><\/math><\/span><span class=\"period\">.<\/span> Also ist <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><\/math> eine obere Schranke von <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">,<\/span> was aber <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> widerspricht. Daher gilt <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\">=<\/mo> <mi>c<\/mi><\/math> und der Satz folgt. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z9875f9fed134\"> <a id=\"x1-96005r59\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.59 <\/span>(Flugreise)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir eine Anwendung des Zwischenwertsatzes aus dem Alltag<\/span> <span class=\"ecti-1095\">pr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sentieren. Angenommen Sie fliegen von Z<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rich nach Lima. Zeigen Sie, dass Sie dabei<\/span> <span class=\"ecti-1095\">den <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quator <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berqueren m<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ssen.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Betrachten Sie Breitengrade als Funktion der Zeit.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Wir wenden uns nun formaleren Anwendungen des Zwischenwertsatzes zu. Der Zwischenwertsatz ist beispielsweise n\u00fctzlich, wenn man versucht Nullstellen von stetigen Funktionen zu finden oder wenn man versucht, \u201eL\u00f6cher\u201c im Bild einer Funktion auszuschliessen. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z8469aa36f9f6\"> <a id=\"x1-96006r60\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.60 <\/span>(Nullstellen von Polynomen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass jedes reelle Polynom von ungeradem Grad eine (reelle) Nullstelle besitzt.<\/span> <span class=\"ecti-1095\">Gehen             Sie             dazu             wie             folgt             vor:             Sei<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">in<\/span> <math display=\"inline\"><mi>\u211d<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">ungerade.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Verwenden Sie den Beweis von Proposition<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/polynome#x1-81008r15\"><span class=\"ecti-1095\">3.15<\/span><\/a> <span class=\"ecti-1095\">und zeigen Sie, dass es Zahlen<\/span> <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">geben muss 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\">&lt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z948210b9a4ad\"> <a id=\"x1-96007r61\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.61 <\/span>(Injektivit\u00e4t und Monotonie)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span> <math display=\"inline\"><mi>I<\/mi><\/math> <span class=\"ecti-1095\">ein                           nicht-leeres                           Intervall                           und<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>I<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine stetige,          injektive          Abbildung.          Zeigen          Sie,          dass<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">streng monoton ist.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Betrachten Sie <\/span><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>I<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">und nehmen Sie zuerst an, dass <\/span><span class=\"maperiod\"><math display=\"inline\"><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\">Was kann man <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber den Wert von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">an einem Punkt <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>I<\/mi><\/math> <span class=\"ecti-1095\">sagen?<\/span><\/p><\/details>  <\/div> <p class=\"indent\">In den folgenden zwei \u00dcbungen m\u00f6chten wir auf einen weiteren Begriff hinweisen, der eng in Verbindung zum Zwischenwertsatz steht. Im zweiten Semester werden wir auf diesen Zusammenhang zur\u00fcckkehren. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zfb586e4ec100\"> <a id=\"x1-96008r62\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.62 <\/span>(Zusammenh\u00e4ngende Teilmengen von <math display=\"inline\"><mi>\u211d<\/mi><\/math>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir nennen eine Teilmenge <\/span><math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecbi-1095\">zusammenh<\/span><span class=\"ecbi-1095\">\u00e4<\/span><span class=\"ecbi-1095\">ngend<\/span><span class=\"ecti-1095\">, wenn es keine zwei offenen Mengen<\/span> <math display=\"inline\"><mi>U<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>V<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>U<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mi>M<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2294<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>V<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mi>M<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>M<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>U<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mi>M<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mi>\u2205<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>V<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mi>M<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mi>\u2205<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-96009r7\" \/><mstyle class=\"maketag\"><mtext>(3.7)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">gibt. Intuitiv ist die Menge <\/span><math display=\"inline\"><mi>M<\/mi><\/math> <span class=\"ecti-1095\">also zusammenh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngend, wenn sie sich nicht durch offene Mengen auseinanderreissen<\/span> <span class=\"ecti-1095\">l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst. Ziel dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung ist es zu zeigen, dass die zusammenh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngenden Teilmengen von<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">gerade<\/span> <span class=\"ecti-1095\">die Intervalle sind.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Teilmenge, die nicht ein Intervall ist. Zeigen Sie unter Verwendung von <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-70003r79\"><span class=\"ecti-1095\">2.79<\/span><\/a><span class=\"ecti-1095\">,<\/span> <span class=\"ecti-1095\">dass <\/span><math display=\"inline\"><mi>M<\/mi><\/math> <span class=\"ecti-1095\">nicht zusammenh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngend ist.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Sei nun <\/span><math display=\"inline\"><mi>I<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein nicht-leeres Intervall und <\/span><math display=\"inline\"><mi>U<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>V<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">offen wie in Gleichung<\/span> (<a href=\"..\/..\/chapter\/der-zwischenwertsatz#x1-96009r7\">3.7<\/a>) <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>I<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>U<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mi>I<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>v<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mi>I<\/mi><\/math> <span class=\"ecti-1095\">und ohne Beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkung der Allgemeinheit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>v<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Betrachten Sie die Menge <\/span><math display=\"inline\"><mi>S<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>s<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>U<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">und zeigen Sie in Analogie zum Beweis des Zwischenwertsatzes, dass das Supremum von<\/span> <math display=\"inline\"><mi>S<\/mi><\/math> <span class=\"ecti-1095\">weder in <\/span><math display=\"inline\"><mi>U<\/mi><\/math> <span class=\"ecti-1095\">noch in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>V<\/mi> <\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">aber in <\/span><math display=\"inline\"><mi>I<\/mi><\/math> <span class=\"ecti-1095\">liegen muss.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z5ac8148ddad2\"> <a id=\"x1-96012r63\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.63 <\/span>(Zwischenwertsatz via Zusammenhang)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie den Zwischenwertsatz in folgenden Schritten. Sei<\/span> <math display=\"inline\"><mi>I<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">ein Intervall<\/span> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>I<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">stetig.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Charakterisierung von Stetigkeit) Zeigen Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede offene Menge <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>U<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">dass <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>U<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">von der Form <\/span><math display=\"inline\"><msup><mrow><mi>U<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-bin\">\u2229<\/mo> <mi>I<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine offene Menge <\/span><math display=\"inline\"><msup><mrow><mi>U<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist.<\/span> <p class=\"noindent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Vergleichen Sie mit <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/stetigkeit#x1-94015r56\"><span class=\"ecti-1095\">3.56<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/p><\/details> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass das Bild von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">zusammenh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngend ist.<\/span> <p class=\"noindent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Nehmen sie an, dass offene Teilmengen <\/span><math display=\"inline\"><mi>U<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>V<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">wie in Gleichung<\/span> (<a href=\"..\/..\/chapter\/der-zwischenwertsatz#x1-96009r7\">3.7<\/a>) <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>I<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">existieren und betrachten Sie deren Urbild unter <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Schliessen Sie auf den Zwischenwertsatz unter Verwendung von <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/der-zwischenwertsatz#x1-96008r62\"><span class=\"ecti-1095\">3.62<\/span><\/a> <span class=\"ecti-1095\">und (ii).<\/span> <p class=\"noindent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>I<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist ein Intervall.<\/span><\/p><\/details><\/dd><\/dl> <\/div> <a id=\"x1-96016r96\"><\/a> \n","rendered":"\n<style scoped=\"scoped\">.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; border-left: 0; border-right: 0; }\ncenter { margin-top:1em; margin-bottom:1em; }\ntd center { margin-top:0em; margin-bottom:0em; }\n.Canvas { position:relative; }\nmath { text-indent: 0em; }\nli p.indent { text-indent: 0em }\nli p:first-child{ margin-top:0em; }\nli p:last-child, li div:last-child { margin-bottom:0.5em; }\nli p~ul:last-child, li p~ol:last-child{ margin-bottom:0.5em; }\n.enumerate1 {list-style-type:decimal;}\n.enumerate2 {list-style-type:lower-alpha;}\n.enumerate3 {list-style-type:lower-roman;}\n.enumerate4 {list-style-type:upper-alpha;}\n.obeylines-h,.obeylines-v {white-space: nowrap; }\ndiv.obeylines-v p { margin-top:0; margin-bottom:0; }\n.overline{ text-decoration:overline; }\n.overline img{ border-top: 1px solid black; }\ntd.displaylines {text-align:center; white-space:nowrap;}\n.centerline {text-align:center;}\n.rightline {text-align:right;}\npre.verbatim {font-family: monospace,monospace; text-align:left; clear:both; }\n.fbox {padding-left:3.0pt; padding-right:3.0pt; text-indent:0pt; border:solid black 0.4pt; }\ndiv.fbox {display:table}\ndiv.center div.fbox {text-align:center; clear:both; padding-left:3.0pt; padding-right:3.0pt; text-indent:0pt; border:solid black 0.4pt; }\ndiv.minipage{width:100%;}\ndiv.center, div.center div.center {text-align: center; margin-left:1em; margin-right:1em;}\ndiv.center {text-align: left;}\ndiv.flushright, div.flushright div.flushright {text-align: right;}\ndiv.flushright div {text-align: left;}\ndiv.flushleft {text-align: left;}\n.underline{ text-decoration:underline; }\n.underline img{ border-bottom: 1px solid black; margin-bottom:1pt; }\n.framebox-c, .framebox-l, .framebox-r { padding-left:3.0pt; padding-right:3.0pt; text-indent:0pt; border:solid black 0.4pt; }\n.framebox-c {text-align:center;}\n.framebox-l {text-align:left;}\n.framebox-r {text-align:right;}\nspan.thank-mark{ vertical-align: super }\nspan.footnote-mark sup.textsuperscript, span.footnote-mark a sup.textsuperscript{ font-size:80%; }\ndiv.tabular, div.center div.tabular {text-align: center; margin-top:0.5em; margin-bottom:0.5em; }\ntable.tabular td p{margin-top:0em;}\ntable.tabular {margin-left: auto; margin-right: auto;}\ntd p:first-child{ margin-top:0em; }\ntd p:last-child{ margin-bottom:0em; }\ndiv.td00{ margin-left:0pt; margin-right:0pt; }\ndiv.td01{ margin-left:0pt; margin-right:5pt; }\ndiv.td10{ margin-left:5pt; margin-right:0pt; }\ndiv.td11{ margin-left:5pt; margin-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\ntd.td00{ padding-left:0pt; padding-right:0pt; }\ntd.td01{ padding-left:0pt; padding-right:5pt; }\ntd.td10{ padding-left:5pt; padding-right:0pt; }\ntd.td11{ padding-left:5pt; padding-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\n.hline hr, .cline hr{ height : 0px; margin:0px; }\n.hline td, .cline td{ padding: 0; }\n.hline hr, .cline hr{border:none;border-top:1px solid black;}\n.tabbing-right {text-align:right;}\ndiv.float, div.figure {margin-left: auto; margin-right: auto;}\ndiv.float img {text-align:center;}\ndiv.figure img {text-align:center;}\n.marginpar,.reversemarginpar {width:20%; float:right; text-align:left; margin-left:auto; margin-top:0.5em; font-size:85%; text-decoration:underline;}\n.marginpar p,.reversemarginpar p{margin-top:0.4em; margin-bottom:0.4em;}\n.reversemarginpar{float:left;}\n.equation td{text-align:center; vertical-align:middle; }\ntd.eq-no{ width:5%; }\ntable.equation { width:100%; }\ndiv.math-display, div.par-math-display{text-align:center;}\nmtr.hline mtd{ border-bottom:black solid 1px; padding-top:2px; padding-bottom:0em; }\nmtr.hline mtd mo{ display:none }\nmath .texttt { font-family: monospace; }\nmath .textit { font-style: italic; }\nmath .textsl { font-style: oblique; }\nmath .textsf { font-family: sans-serif; }\nmath .textbf { font-weight: bold; }\nmo.MathClass-op + mi{margin-left:0.3em}\nmi + mo.MathClass-op{margin-left:0.3em}\n math mstyle[mathvariant=\"bold\"] { font-weight: bold; font-style: normal; }\n math mstyle[mathvariant=\"normal\"] { font-weight: normal; font-style: normal; }\n.partToc a, .partToc, .likepartToc a, .likepartToc {line-height: 200%; font-weight:bold; font-size:110%;}\n.index-item, .index-subitem, .index-subsubitem {display:block}\ndiv.caption {text-indent:-2em; margin-left:3em; margin-right:1em; text-align:left;}\ndiv.caption span.id{font-weight: bold; white-space: nowrap; }\nh1.partHead{text-align: center}\np.bibitem { text-indent: -2em; margin-left: 2em; margin-top:0.6em; margin-bottom:0.6em; }\np.bibitem-p { text-indent: 0em; margin-left: 2em; margin-top:0.6em; margin-bottom:0.6em; }\n.paragraphHead, .likeparagraphHead { margin-top:2em; font-weight: bold;}\n.subparagraphHead, .likesubparagraphHead { font-weight: bold;}\n.quote {margin-bottom:0.25em; margin-top:0.25em; margin-left:1em; margin-right:1em; text-align:justify;}\n.verse{white-space:nowrap; margin-left:2em}\ndiv.maketitle {text-align:center;}\nh2.titleHead{text-align:center;}\ndiv.maketitle{ margin-bottom: 2em; }\ndiv.author, div.date {text-align:center;}\ndiv.thanks{text-align:left; margin-left:10%; font-size:85%; font-style:italic; }\ndiv.author{white-space: nowrap;}\n.quotation {margin-bottom:0.25em; margin-top:0.25em; margin-left:1em; }\n.abstract p {margin-left:5%; margin-right:5%;}\ndiv.abstract {width:100%;}\ndiv.tabular, div.center div.tabular {text-align: center; margin-top:0.5em; margin-bottom:0.5em; }\ntable.tabular td p{margin-top:0em;}\ntable.tabular {margin-left: auto; margin-right: auto;}\ntd p:first-child{ margin-top:0em; }\ntd p:last-child{ margin-bottom:0em; }\ndiv.td00{ margin-left:0pt; margin-right:0pt; }\ndiv.td01{ margin-left:0pt; margin-right:5pt; }\ndiv.td10{ margin-left:5pt; margin-right:0pt; }\ndiv.td11{ margin-left:5pt; margin-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\ntd.td00{ padding-left:0pt; padding-right:0pt; }\ntd.td01{ padding-left:0pt; padding-right:5pt; }\ntd.td10{ padding-left:5pt; padding-right:0pt; }\ntd.td11{ padding-left:5pt; padding-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\n.hline hr, .cline hr{ height : 0px; margin:0px; }\n.hline td, .cline td{ padding: 0; }\n.hline hr, .cline hr{border:none;border-top:1px solid black;}\n.equation-star td{text-align:center; vertical-align:middle; }\ntable.equation-star { width:100%; border-bottom-color: rgb(255,255,255); }\n#content table.equation-star, #content table.equation-star tbody tr td { border: 0px none rgb(255,255,255); }\nmtd.align-odd{margin-left:2em; text-align:right;}\nmtd.align-even{margin-right:2em; text-align:left;}\n.boxed{border: 1px solid black; padding-left:2px; padding-right:2px;}\n.rotatebox{display: inline-block;}\n.item-head{float:left;width:2em;clear:left;}\n.item-content{margin-left:2em;}\n .foreignobject {line-height:100%; font-size:120%; font-family:STIXgeneral,Times,Symbol,cmr10,CMSY10,CMEX10;padding:0; margin:0; text-align:center; }\nmath {vertical-align:baseline; line-height:100%; font-size:100%; font-family:STIXGeneral,Times,Symbol, cmr10,cmsy10,cmex10,cmmi10; font-style: normal; margin:0; padding:0; }\n\n.entry-title{display: none}\n\ndiv.newtheorem { margin-bottom: 2em; margin-top: 2em; border: 1px solid #333; background: #c7e4da; border-color: #4eb79e;}\ndiv.newtheorem h3 { background: #4eb79e; color: white; padding: 0px 15px 0px 15px; margin-top: 12px}\ndiv.newtheorem p { padding: 15px 15px 15px 15px; }\n\ndiv.newtheorem p span.head .ecbx-1095{font-weight: bold}\ndiv.newtheorem p .ecti-1095{font-style: italic}\ndiv.newtheorem div.custom-itemize{font-style: italic}\ndiv.quote{font-style: italic}\ndiv.newtheorem dl, dl.enumerate {display: grid; grid-template-columns: 5% auto; align-items: start; margin-top: 1em}\ndiv.newtheorem dl dd, dl.enumerate dd {margin-bottom: 0.5em}\ndiv.newtheorem dl dt, dl.enumerate dt {font-weight: normal; margin-top: 0px; text-align: right; margin-right: 15%}\ndiv.newtheorem dl dd {font-style: italic}\ndiv.newtheorem dl dt {font-style: italic}\ndiv.proof p span.ecti-1095 {font-style: italic}\ndiv.figure p img { margin-left: auto; margin-right: auto; display: block; }\ndiv.mefigcentered, div.figure { text-align: center }\n\ndl:after {content:\"\";display:table;clear:both;}\ndd {padding:.5em 0;}\ndl {width:100%;}\ndt, dd {display:inline-block; width:125%;}\ndt {text-align:right; font-weight:bold; clear:left; float:left;}\ndd {width:100%; padding-left:1em; padding-top: 0px; clear:right;}\ndd + dd {float:right; clear:both;}\ndd + dt {clear:both;}\ndt + dt {width: 100%; float: none; padding: 0 70% 0 0;}\ndt + dt + dd {margin-top: -2em;}\ndt + dt + dd + dt {margin-top: 2em;}\n<\/style>\n<style scoped=\"scoped\">\n\/* CSS Analysis-Skript D-Math ETHZ *\/\n\n\/* Uniform Font, also for headers *\/\nh3 {\n\tfont-family: \"Times New Roman\", serif;\n\tmargin-bottom: 35px;\n}\nh4 {\n\tfont-family: \"Times New Roman\", serif;\n}\nh5 {\n\tfont-family: \"Times New Roman\", serif;\n}\n\n\/* Bold font, e.g. for 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=\"z8d303c0fcb9f\" class=\"sectionHead\"><span class=\"titlemark\">3.6 <\/span> <a id=\"x1-960006\"><\/a>Der Zwischenwertsatz<\/h3> <p class=\"noindent\">In diesem Abschnitt wollen wir einen fundamentalen Satz beweisen, der die Heuristik, dass der Graph einer stetigen Funktion auf einem Intervall \u201eeine durchgehende Kurve\u201c darstellt, formalisiert. Wir sagen, dass eine reelle Zahl <math display=\"inline\"><mi>c<\/mi><\/math> <span class=\"ecbx-1095\">zwischen zwei<\/span> <span class=\"ecbx-1095\">reellen Zahlen <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> <span class=\"ecbx-1095\">liegt<\/span>, falls <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>c<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> oder <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>c<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> gilt. Wir sagen&nbsp;<math display=\"inline\"><mi>c<\/mi><\/math> liegt <span class=\"ecbx-1095\">echt<\/span> <span class=\"ecbx-1095\">zwischen<\/span><span class=\"ecbx-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecbx-1095\">und<\/span><span class=\"ecbx-1095\">&nbsp;<\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <\/math> falls&nbsp;<math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>c<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><\/math> oder <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>c<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/math> ist. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zd703df118694\"> <a id=\"x1-96001r58\"><\/a> <span class=\"ecbx-1095\">Satz 3.58 <\/span>(Zwischenwertsatz)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>I<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein Intervall, <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>I<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine stetige Funktion und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>I<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes <\/span><math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">zwischen <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gibt es ein <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">zwischen <\/span><math display=\"inline\"><mi>a<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>b<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass <\/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>c<\/mi><\/math> <span class=\"ecti-1095\">gilt.<\/span> <\/p> <\/div> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=575&amp;url=Pictures\/funktionen_R\/zwsatz\/zwsatz.pdf\"><img decoding=\"async\" id=\"z89e665fb5463\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/funktionen_R\/zwsatz\/zwsatz.svg\" width=\"575\" \/><\/div> <a id=\"x1-96002r6\"><\/a> <a id=\"x1-96003\"><\/a> <br \/><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;3.6:     <\/span><span class=\"content\">Der     Graph     einer     stetigen     Funktion     kann               keine Spr\u00fcnge machen und die Funktion nimmt alle Werte zwischen               <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math>               und               <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math>               an.                                                                                          &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <p class=\"indent\">Wie wir sehen werden, verwendet der Beweis die Existenz des Supremums (und damit das Vollst\u00e4ndigkeitsaxiom). <\/p><div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir nehmen ohne Beschr\u00e4nkung der Allgemeinheit an, dass <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> und <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> gilt (falls <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> ist, betrachtet man zuerst <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><\/math> und bemerkt, dass die Aussage des Satzes f\u00fcr <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><\/math> die Aussage des Satzes f\u00fcr <math display=\"inline\"><mi>f<\/mi><\/math> impliziert). <\/p><p class=\"indent\">Sei nun <span class=\"maperiod\"><math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> Falls <math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> oder <math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> gilt, sind wir fertig. Also angenommen <span class=\"maperiod\"><math display=\"inline\"><mi>c<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Wir definieren <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><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\">\u2264<\/mo> <mi>c<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und bemerken, dass <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>X<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">,<\/span>                                                                                                                                                                           wodurch <math display=\"inline\"><mi>X<\/mi><\/math> nicht-leer und von oben beschr\u00e4nkt ist. Nach Satz <a href=\"..\/..\/chapter\/maximum-und-supremum#x1-64002r59\">2.59<\/a> existiert daher <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> Wir werden nun die Stetigkeit von <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> verwenden, um zu zeigen, dass <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>c<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">F\u00fcr jedes <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> gibt es ein <span class=\"maperiod\"><math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> so dass f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> gilt <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\u03b4<\/mi><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">\u21d2<\/mo><mspace class=\"thickpace\" width=\"0.28em\" \/><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">|<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>\ud835\udf00<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-96004r6\" \/><mstyle class=\"maketag\"><mtext>(3.6)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Angenommen <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>c<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann folgt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> wegen <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>c<\/mi><\/math> und <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> Wir wenden nun die Stetigkeit von <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> an und finden f\u00fcr <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>c<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> ein <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> wie in (<a href=\"..\/..\/chapter\/der-zwischenwertsatz#x1-96004r6\">3.6<\/a>). Da <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> ist, existiert ein <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> F\u00fcr dieses <math display=\"inline\"><mi>x<\/mi><\/math> gilt dann                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><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><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><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> <mo class=\"MathClass-bin\">+<\/mo> <mi>c<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>c<\/mi><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Also muss <math display=\"inline\"><mi>x<\/mi><\/math> in <math display=\"inline\"><mi>X<\/mi><\/math> liegen, was aber <math display=\"inline\"><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>x<\/mi><\/math> widerspricht. <\/p><p class=\"indent\">Angenommen <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>c<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann folgt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>a<\/mi><\/math> wegen <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>c<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wir verwenden wieder die Stetigkeit von <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> und finden zu <math display=\"inline\"><mi>\ud835\udf00<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>c<\/mi><\/math> ein <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> mit der Eigenschaft in (<a href=\"..\/..\/chapter\/der-zwischenwertsatz#x1-96004r6\">3.6<\/a>) . F\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> gilt dadurch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><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><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>c<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wodurch <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u2209<\/mo> <mi>X<\/mi><\/math> und daher <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-bin\">\u2229<\/mo> <mi>X<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u2205<\/mi><\/math><\/span><span class=\"period\">.<\/span> Also ist <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><\/math> eine obere Schranke von <span class=\"maperiod\"><math display=\"inline\"><mi>X<\/mi><\/math><\/span><span class=\"period\">,<\/span> was aber <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sup<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>X<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> widerspricht. Daher gilt <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\">=<\/mo> <mi>c<\/mi><\/math> und der Satz folgt. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z9875f9fed134\"> <a id=\"x1-96005r59\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.59 <\/span>(Flugreise)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">In dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung m<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">chten wir eine Anwendung des Zwischenwertsatzes aus dem Alltag<\/span> <span class=\"ecti-1095\">pr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sentieren. Angenommen Sie fliegen von Z<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">rich nach Lima. Zeigen Sie, dass Sie dabei<\/span> <span class=\"ecti-1095\">den <\/span><span class=\"ecti-1095\">\u00c4<\/span><span class=\"ecti-1095\">quator <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berqueren m<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ssen.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Betrachten Sie Breitengrade als Funktion der Zeit.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Wir wenden uns nun formaleren Anwendungen des Zwischenwertsatzes zu. Der Zwischenwertsatz ist beispielsweise n\u00fctzlich, wenn man versucht Nullstellen von stetigen Funktionen zu finden oder wenn man versucht, \u201eL\u00f6cher\u201c im Bild einer Funktion auszuschliessen. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z8469aa36f9f6\"> <a id=\"x1-96006r60\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.60 <\/span>(Nullstellen von Polynomen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie, dass jedes reelle Polynom von ungeradem Grad eine (reelle) Nullstelle besitzt.<\/span> <span class=\"ecti-1095\">Gehen             Sie             dazu             wie             folgt             vor:             Sei<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">=<\/mo><msubsup><mrow><mi class=\"MathClass-op\"> \u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>j<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msubsup><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msub><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>j<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">in<\/span> <math display=\"inline\"><mi>\u211d<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">mit<\/span> <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><mi>n<\/mi><\/math> <span class=\"ecti-1095\">ungerade.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Verwenden Sie den Beweis von Proposition<\/span><span class=\"ecti-1095\">&nbsp;<\/span><a href=\"..\/..\/chapter\/polynome#x1-81008r15\"><span class=\"ecti-1095\">3.15<\/span><\/a> <span class=\"ecti-1095\">und zeigen Sie, dass es Zahlen<\/span> <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">geben muss 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\">&lt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z948210b9a4ad\"> <a id=\"x1-96007r61\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.61 <\/span>(Injektivit\u00e4t und Monotonie)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei<\/span> <math display=\"inline\"><mi>I<\/mi><\/math> <span class=\"ecti-1095\">ein                           nicht-leeres                           Intervall                           und<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>I<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine stetige,          injektive          Abbildung.          Zeigen          Sie,          dass<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">streng monoton ist.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Betrachten Sie <\/span><math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>I<\/mi><\/math> <span class=\"ecti-1095\">mit <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> <span class=\"ecti-1095\">und nehmen Sie zuerst an, dass <\/span><span class=\"maperiod\"><math display=\"inline\"><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\">Was kann man <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ber den Wert von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">an einem Punkt <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>I<\/mi><\/math> <span class=\"ecti-1095\">sagen?<\/span><\/p><\/details>  <\/div> <p class=\"indent\">In den folgenden zwei \u00dcbungen m\u00f6chten wir auf einen weiteren Begriff hinweisen, der eng in Verbindung zum Zwischenwertsatz steht. Im zweiten Semester werden wir auf diesen Zusammenhang zur\u00fcckkehren. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zfb586e4ec100\"> <a id=\"x1-96008r62\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.62 <\/span>(Zusammenh\u00e4ngende Teilmengen von <math display=\"inline\"><mi>\u211d<\/mi><\/math>)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir nennen eine Teilmenge <\/span><math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecbi-1095\">zusammenh<\/span><span class=\"ecbi-1095\">\u00e4<\/span><span class=\"ecbi-1095\">ngend<\/span><span class=\"ecti-1095\">, wenn es keine zwei offenen Mengen<\/span> <math display=\"inline\"><mi>U<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>V<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">mit<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>U<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mi>M<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2294<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>V<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mi>M<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>M<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>U<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mi>M<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mi>\u2205<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>V<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mi>M<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mi>\u2205<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-96009r7\" \/><mstyle class=\"maketag\"><mtext>(3.7)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">gibt. Intuitiv ist die Menge <\/span><math display=\"inline\"><mi>M<\/mi><\/math> <span class=\"ecti-1095\">also zusammenh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngend, wenn sie sich nicht durch offene Mengen auseinanderreissen<\/span> <span class=\"ecti-1095\">l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst. Ziel dieser <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung ist es zu zeigen, dass die zusammenh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngenden Teilmengen von<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">gerade<\/span> <span class=\"ecti-1095\">die Intervalle sind.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Teilmenge, die nicht ein Intervall ist. Zeigen Sie unter Verwendung von <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/erste-konsequenzen-der-vollstaendigkeit#x1-70003r79\"><span class=\"ecti-1095\">2.79<\/span><\/a><span class=\"ecti-1095\">,<\/span> <span class=\"ecti-1095\">dass <\/span><math display=\"inline\"><mi>M<\/mi><\/math> <span class=\"ecti-1095\">nicht zusammenh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngend ist.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Sei nun <\/span><math display=\"inline\"><mi>I<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein nicht-leeres Intervall und <\/span><math display=\"inline\"><mi>U<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>V<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">offen wie in Gleichung<\/span> (<a href=\"..\/..\/chapter\/der-zwischenwertsatz#x1-96009r7\">3.7<\/a>) <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>I<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>U<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mi>I<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>v<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>V<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mi>I<\/mi><\/math> <span class=\"ecti-1095\">und ohne Beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkung der Allgemeinheit <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>u<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>v<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Betrachten Sie die Menge <\/span><math display=\"inline\"><mi>S<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>s<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mo class=\"MathClass-open\">[<\/mo><mi>u<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>s<\/mi><mo class=\"MathClass-close\">]<\/mo> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>U<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">und zeigen Sie in Analogie zum Beweis des Zwischenwertsatzes, dass das Supremum von<\/span> <math display=\"inline\"><mi>S<\/mi><\/math> <span class=\"ecti-1095\">weder in <\/span><math display=\"inline\"><mi>U<\/mi><\/math> <span class=\"ecti-1095\">noch in <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>V<\/mi> <\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">aber in <\/span><math display=\"inline\"><mi>I<\/mi><\/math> <span class=\"ecti-1095\">liegen muss.<\/span><\/dd><\/dl> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z5ac8148ddad2\"> <a id=\"x1-96012r63\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 3.63 <\/span>(Zwischenwertsatz via Zusammenhang)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie den Zwischenwertsatz in folgenden Schritten. Sei<\/span> <math display=\"inline\"><mi>I<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">[<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">ein Intervall<\/span> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>I<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">stetig.<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">(Charakterisierung von Stetigkeit) Zeigen Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jede offene Menge <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>U<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">dass <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>U<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">von der Form <\/span><math display=\"inline\"><msup><mrow><mi>U<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-bin\">\u2229<\/mo> <mi>I<\/mi><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine offene Menge <\/span><math display=\"inline\"><msup><mrow><mi>U<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist.<\/span> <div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Vergleichen Sie mit <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/stetigkeit#x1-94015r56\"><span class=\"ecti-1095\">3.56<\/span><\/a><span class=\"ecti-1095\">.<\/span><\/p><\/details> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass das Bild von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">zusammenh<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ngend ist.<\/span> <div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><span class=\"ecti-1095\">Nehmen sie an, dass offene Teilmengen <\/span><math display=\"inline\"><mi>U<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>V<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">wie in Gleichung<\/span> (<a href=\"..\/..\/chapter\/der-zwischenwertsatz#x1-96009r7\">3.7<\/a>) <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>M<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>I<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">existieren und betrachten Sie deren Urbild unter <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Schliessen Sie auf den Zwischenwertsatz unter Verwendung von <\/span><span class=\"ecti-1095\">\u00dc<\/span><span class=\"ecti-1095\">bung <\/span><a href=\"..\/..\/chapter\/der-zwischenwertsatz#x1-96008r62\"><span class=\"ecti-1095\">3.62<\/span><\/a> <span class=\"ecti-1095\">und (ii).<\/span> <div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>I<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist ein Intervall.<\/span><\/p><\/details><\/dd><\/dl> <\/div> <a id=\"x1-96016r96\"><\/a> \n","protected":false},"author":1089,"menu_order":6,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-47","chapter","type-chapter","status-publish","hentry"],"part":41,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/47","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/users\/1089"}],"version-history":[{"count":0,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/47\/revisions"}],"part":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/parts\/41"}],"metadata":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/47\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=47"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=47"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=47"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=47"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}