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zI6rC{nqO!5UTEsUXn?Wzp^E^;16)6XlMMGvS3h`>&iDHo-^DsWrkZr7x{}jp?v~FeCp5|_H?w*$ z9`!B6$1)80E&S@Y`+Sl{i?o6F*ZMhB3)Fe}(FeHG4_+`jW@f~N;0fdTeof?iot&JS zxSTq?OzldUx-wBM`})F&)~rc%SYf4E5vb7L)FYzM-z@c00bQCVmX9)98_IJ)c)20c zgplurOs@u(37J$c*l^|hqw;xssK)umJ!2>Hp+59A^T}VfHCFmL%N-a*ZkyrT9{zKm zdC%49@`Ec)Duz@REv4c#mu!6qO|JyWzDp-dia=5S^b`US<4yEq?0.$$ $f$ stetig in $a \implies$ für dieses $\epsilon \; \exists \delta > 0$ s.d. für $K_\delta(a) \cap D$ gilt $f(K_\delta(a)\cap D) \subset (K_\epsilon(f(a))\cap f(D))\subset O$ $\implies \forall a \in D$ (mit $f(a)\in O$) gilt $$(K_\delta(a) \cap D) \subset f^{-1}(O) \implies f^{-1}(O)$$ relativ-offen. + \item Sei $(x^{(k)}_{k\in \N})$ eine konvergente Folge in $f^{-1}(A)$ und $\lim\limits_{k\to\infty}x^{(k)} = x \in \overline{D}$. Aufgrund der Stetigkeit von $f$ konvergiert die Bildfolge $\left(f\left(x^{(k)}\right)\right)_{k\in \N}$ und $\lim\limits_{k\to\infty}\underbrace{f\left(x^{(k)}\right)}_{\in A} = f(x)$. Da $A$ abgeschlossen ist, gilt wegen der Charakterisierung über Folgenkonvergenz auch $f(x) \in A \implies x \in f^{-1}(A) \implies f^{-1}(A)$ abgeschlossen. + \item \textbf{Z.Z} $f(K) \subset \K$ beschränkt und abgeschlossen $(\implies$ kompakt). + \begin{itemize} + \item Die Beschränktheit folgt aus der Beschränktheit von stetigen Funktionen auf kompakten Mengen. + \item Abgeschlossenheit: Sei $\left(y^{(k)}\right)_{k\in \N} \subset f(K)$ eine beliebige Folge mit $\lim\limits_{k\to\infty}y^{(k)} = y \in \K$. Die Urbildfolge $\left(x^{(k)}\right)_{k\in \N} \bigg(= \underbrace{f^{-1}\left(y^{(k)}\right)}_{\text{Urbild von }y^{(k)}}\bigg)$ in $K$ hat aufgrund der Kompaktheit von $K$ eine konvergente Teilfolge $\left(x^{(k_j)}\right)_{j\in \N}$ mit $\lim\limits_{j\to\infty}x^{(kj)} = x\in K$. Wegen der Stetigkeit von $f$ ist $f(x) = \lim\limits_{j\to\infty}\underbrace{f\left(x^{(k_j)}\right)}_{=y^{k_j}} = y \implies y \in f(K) \implies f(K)$ abgeschlossen. + \end{itemize} + \item Ann: $f(G)$ nicht zusammenhängend. Nach Definition existieren dann $U,V\in \K,\; U\neq \emptyset, V\neq \emptyset,\; U\cap V = \emptyset$, relativ offen, s.d. $f(G) = U \cup V$. Für die Urbildmengen gilt also $$U' = \{x \in G|f(x)\in U\},\; V' = \{x \in G|f(x)\in V\},\; U'\cap V' = \emptyset, U' \neq \emptyset, V'\neq \emptyset,$$ nach 1) sind $U'$ und $V'$ relativ offen und $G = U' \cup V'$ + \begin{align*} + &\implies G\text{ nicht zusammenhängend \lightning}\\ + &\implies f(G) \text{ zusammenhängend.} + \end{align*} + \end{enumerate} + \end{proof} +\begin{satz} + Sei $f: D\subset \K^n \to \R$ stetig und $D$ zusammenhängend. Dann nimmt $f$ für alle $a, b \in D$ jeden Wert zwischen $f(a)$ und $f(b)$ an. Insbesondere hat $f$ eine Nullstelle in $D$, falls $f(a)\cdot f(b) < 0$. +\end{satz} +\begin{proof} + Wegen Lemma \ref{lemma:stetigesurbild} ist der Bildbereich $f(D)\subset \R$ zusammenhängend.\\ + \textbf{Z.Z.} $f(D)$ ist ein (zusammenhängendes) Intervall.\\ + Annahme: $f(D)$ ist kein Intervall $\implies \exists f(x), f(y) \in f(D)$ und $z\notin f(D)$ zwischen $f(x)$ und $f(y)$. Dann sind die Mengen $$U\coloneqq f(D)\cap (-\infty, z),\; V\coloneqq f(D)\cap (z, \infty)$$ disjunkt, $U\cap V = \emptyset,\; U \neq \emptyset,\; V\neq \emptyset$, $U, V$ sind relativ-offen bzgl. $f(D)$ und $U\cap V = f(D)$ $$\implies f(D)\text{ nicht zusammenhängend \lightning.}$$ +\end{proof} +\section{Vektor- und Matrix-wertige Funktionen} +\begin{align*} + f:&D \subset \K^n \to \K^m\\ + f:&D \subset \K^n \to \K^{r\times s}\\ + f:&D \subset \K^{m\times n} \to \K^{r\times s} +\end{align*} +\begin{itemize} + \item Mit Hilfe von Normen in $\K^n$ und $K^{m\times n}$ sind die Definitionen von Stetigkeit etc. übertragbart auf Vektor- und Matrix-wertige Funktionen. $\implies$ solche stetigen Abbildungen auf kompakten Mengen sind gleichmäßig stetig und beschränkt. + \item $f: D\subset \K^n \to \K^m$ stetig $\Leftrightarrow$ alle Komponenten $f_i:D\subset \K^n \to \K,\; i= 1, \dots, m$ sind stetig (genauso für $f: D\subset \K^{n\times m} \to \K^{r\times s}$ etc.) +\end{itemize} +\begin{lemma} + Seien $g: D\subset \K^n\to B \subset \K^m$ und $f:B\to \K^r$ stetig. Dann ist die Komposition $f\circ g: D\subset \K^n \to \K^r$ stetig. +\end{lemma} +\begin{proof} + Sei $x\in D, x^{(k)} \in D$ s.d. $\lim\limits_{k\to\infty}x^{(k)} = x$. Es gilt + \begin{align*} + g\text{ stetig }&\implies y^{(k)} = g(x^{(k)}) \underset{k\to \infty}{\to} g(x) =: y\\ + f\text{ stetig }&\implies (f\circ g)\left(x^{(k)}\right) = f\left(g\left(x^{(k)}\right)\right) = f(y^{(k)}) \underset{k\to \infty}{\to} f(y) + \end{align*} + Also ist $(f\circ g)$ stetig. +\end{proof} +\begin{lemma} + Sei $D\subset \K^n$ kompakt und $f: D\to B\subset \K^n$ injektiv und stetig. Dann ist die Umkehrfunktion $f^{-1}: B\to D$ stetig. +\end{lemma} +\begin{proof} + Sei $\left(y^{(k)}\right)_{k\in \N}$ eine Folge in $B$ mit $\lim\limits_{k\to\infty}y^{(k)} = y \in B$.\\ \textbf{Z.Z.} $\underbrace{f^{-1}\left(y^{(k)}\right)}_{=: x^{(k)}} \underset{k\to \infty}{\to} \underbrace{f^{-1}(y)}_{=: x}\; (\implies f^{-1}$ stetig), d.h. $x^{(k)} \underset{k\to \infty}{\to} x$.\\ Die Folge $\left(x^{(k)}\right)\subset D$ ist aufgrund der Beschränktheit von $D$ auch beschränkt, also existiert eine konvergente Teilfolge $\left(x^{(k_j)}\right)_{j\in \N}$ mit $\lim\limits_{j\to\infty}x^{(k_j)} = \xi \in D$. Wegen der Stetigkeit von $f$ ist $f(x^{(k_j)}) \underset{j\to \infty}{\to} f(\xi)$. Außerdem gilt $$f\left(x^{(k_j)}\right) =y^{(k_j)} \underset{j\to \infty}{\to} y \implies y = f(\xi) \overset{y = f(x)}{\underset{f\text{ injektiv}}{\implies}} x = \xi.$$ + Also konvergieren alle konvergenten Teilfolgen von $\left(x^{(k)}\right)$ gegen $x \implies x^{(k)} \underset{k\to \infty}{\to} x$. +\end{proof} +\end{document} \ No newline at end of file diff --git a/analysisII.pdf b/analysisII.pdf index 7cde7fd9642737e349c09a21a068ce5cc1de66b1..26b7449a421ff37628a219519d004ac121f9547e 100644 GIT binary patch delta 164972 zcmYJZV|XRO0lv`r&OekiCgc_;O^cPs zCdsz7_nf{-jIs>2vrqn-V}yfQYfe^33D1cgP}-uq<~jYT%deyNAzx(YAyUXdws7p| z+u82yAbW1i#2o2WP#Q3w`l6?&hmn7cLa2U z%}fdr%g`D{OB^T`NtJPSOB0JeMd3}P8(K-`VxHqTw9u;;noOTwHzXc-Z8EF05^I&W zmYz*jCnFTVY65d$EJ`Pdt`cZm086eVwK)@IY#@UZ%yMO?pRwr=+ieKUF%y3i#j9!K z(u=^ab^F=jc%wcYV~F;?^GaG9b_x<2ja)`E=0>^ygOm^nnE1_GkanMtr>Hq$tfQ5l zCZtbj;Fy)iyY0}05Kmg7W~#stVmtJX8kVsD6N`BI@o|xff?#X@7jR~(6}D*QMG)c4 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z4NM)4U0t1BElf;IoXw5gOkC|0^a!f~S_F*^W24E65nA=Km(2Z~9R(U5&isDDaz@4? zPcuvP$(v7FrgKS2X=GGPU*Ig_ai^ha9#bb*U=gzlmjwgwgZf$i|Fg=MYPz{RIWO3$ z+iIl29vD8aHRwdh#1d<(4UBq64rt{ZwmEW0N9Qop5l5RGM<$g+LT{XeR1R}3anM@g zXq9q+@5}+a6lc5ol$A^N-kM-{qOmlh#qva}?TKa~jew8yZ*F(G!D7G0QBcI0fo~y` zQUGUY0I#V)i@+5p*(--lOdOO=9F4a)Xm4?}E^%1y_2I7D>!8k-wYx&s>8-5_$!l9% z6*_O0.$$ $f$ stetig in $a \implies$ für dieses $\epsilon \; \exists \delta > 0$ s.d. für $K_\delta(a) \cap D$ gilt $f(K_\delta(a)\cap D) \subset (K_\epsilon(f(a))\cap f(D))\subset O$ $\implies \forall a \in D$ (mit $f(a)\in O$) gilt $$(K_\delta(a) \cap D) \subset f^{-1}(O) \implies f^{-1}(O)$$ relativ-offen. - \item Sei $(x^{(k)}_{k\in \N})$ eine konvergente Folge in $f^{-1}(A)$ und $\lim\limits_{k\to\infty}x^{(k)} = x \in \overline{D}$. Aufgrund der Stetigkeit von $f$ konvergiert die Bildfolge $\left(f\left(x^{(k)}\right)\right)_{k\in \N}$ und $\lim\limits_{k\to\infty}\underbrace{f\left(x^{(k)}\right)}_{\in A} = f(x)$. Da $A$ abgeschlossen ist, gilt wegen der Charakterisierung über Folgenkonvergenz auch $f(x) \in A \implies x \in f^{-1}(A) \implies f^{-1}(A)$ abgeschlossen. + \item Sei $O\subset f(D)$ eine relativ-offene Menge. Für $O$ gilt: $\forall f(a) \in O \ \exists$ relative Kugelumgebung in $O$: $$(K_\epsilon(f(a)) \cap f(D)) \subset O\text{ für ein } \epsilon >0.$$ $f$ stetig in $a \implies$ für dieses $\epsilon \; \exists \delta > 0$ s.d. für $K_\delta(a) \cap D$ gilt $f(K_\delta(a)\cap D) \subset (K_\epsilon(f(a))\cap f(D))\subset O$ $\implies \forall a \in D$ (mit $f(a)\in O$) gilt $$(K_\delta(a) \cap D) \subset f^{-1}(O) \implies f^{-1}(O)$$ relativ-offen. + \item Sei $(x^{(k)})_{k\in \N}$ eine konvergente Folge in $f^{-1}(A)$ und $\lim\limits_{k\to\infty}x^{(k)} = x \in \overline{D}$. Aufgrund der Stetigkeit von $f$ konvergiert die Bildfolge $\left(f\left(x^{(k)}\right)\right)_{k\in \N}$ und $\lim\limits_{k\to\infty}\underbrace{f\left(x^{(k)}\right)}_{\in A} = f(x)$. Da $A$ abgeschlossen ist, gilt wegen der Charakterisierung über Folgenkonvergenz auch $f(x) \in A \implies x \in f^{-1}(A) \implies f^{-1}(A)$ abgeschlossen. \item \textbf{Z.Z} $f(K) \subset \K$ beschränkt und abgeschlossen $(\implies$ kompakt). \begin{itemize} \item Die Beschränktheit folgt aus der Beschränktheit von stetigen Funktionen auf kompakten Mengen. - \item Abgeschlossenheit: Sei $\left(y^{(k)}\right)_{k\in \N} \subset f(K)$ eine beliebige Folge mit $\lim\limits_{k\to\infty}y^{(k)} = y \in \K$. Die Urbildfolge $\left(x^{(k)}\right)_{k\in \N} \bigg(= \underbrace{f^{-1}\left(y^{(k)}\right)}_{\text{Urbild von }y^{(k)}}\bigg)$ in $K$ hat aufgrund der Kompaktheit von $K$ eine konvergente Teilfolge $\left(x^{(k_j)}\right)_{j\in \N}$ mit $\lim\limits_{j\to\infty}x^{(kj)} = x\in K$. Wegen der Stetigkeit von $f$ ist $f(x) = \lim\limits_{j\to\infty}\underbrace{f\left(x^{(k_j)}\right)}_{=y^{k_j}} = y \implies y \in f(K) \implies f(K)$ abgeschlossen. + \item Abgeschlossenheit: Sei $\left(y^{(k)}\right)_{k\in \N} \subset f(K)$ eine beliebige Folge mit $\lim\limits_{k\to\infty}y^{(k)} = y \in \K$. Die Urbildfolge $\left(x^{(k)}\right)_{k\in \N} \bigg(= \underbrace{f^{-1}\left(y^{(k)}\right)}_{\text{Urbild von }y^{(k)}}\bigg)$ in $K$ hat aufgrund der Kompaktheit von $K$ eine konvergente Teilfolge $\left(x^{(k_j)}\right)_{j\in \N}$ mit $\lim\limits_{j\to\infty}x^{(k_j)} = x\in K$. Wegen der Stetigkeit von $f$ ist $f(x) = \lim\limits_{j\to\infty}\underbrace{f\left(x^{(k_j)}\right)}_{=y^{k_j}} = y \implies y \in f(K) \implies f(K)$ abgeschlossen. \end{itemize} - \item Ann: $f(G)$ nicht zusammenhängend. Nach Definition existieren dann $U,V\in \K,\; U\neq \emptyset, V\neq \emptyset,\; U\cap V = \emptyset$, relativ offen, s.d. $f(G) = U \cup V$. Für die Urbildmengen gilt also $$U' = \{x \in G|f(x)\in U\},\; V' = \{x \in G|f(x)\in V\},\; U'\cap V' = \emptyset, U' \neq \emptyset, V'\neq \emptyset,$$ nach 1) sind $U'$ und $V'$ relativ offen und $G = U' \cup V'$ + \item Ann: $f(G)$ nicht zusammenhängend. Nach Definition existieren dann $U,V\in \K^n,\; U\neq \emptyset, V\neq \emptyset,\; U\cap V = \emptyset$, relativ offen, s.d. $f(G) = U \cup V$. Für die Urbildmengen gilt also $$U' = \{x \in G|f(x)\in U\},\; V' = \{x \in G|f(x)\in V\},\; U'\cap V' = \emptyset, U' \neq \emptyset, V'\neq \emptyset,$$ nach 1) sind $U'$ und $V'$ relativ offen und $G = U' \cup V'$ \begin{align*} &\implies G\text{ nicht zusammenhängend \lightning}\\ &\implies f(G) \text{ zusammenhängend.} \end{align*} \end{enumerate} \end{proof} -\begin{satz} +\begin{satz}[Zwischenwertsatz] Sei $f: D\subset \K^n \to \R$ stetig und $D$ zusammenhängend. Dann nimmt $f$ für alle $a, b \in D$ jeden Wert zwischen $f(a)$ und $f(b)$ an. Insbesondere hat $f$ eine Nullstelle in $D$, falls $f(a)\cdot f(b) < 0$. \end{satz} \begin{proof} @@ -54,21 +54,21 @@ \textbf{Z.Z.} $f(D)$ ist ein (zusammenhängendes) Intervall.\\ Annahme: $f(D)$ ist kein Intervall $\implies \exists f(x), f(y) \in f(D)$ und $z\notin f(D)$ zwischen $f(x)$ und $f(y)$. Dann sind die Mengen $$U\coloneqq f(D)\cap (-\infty, z),\; V\coloneqq f(D)\cap (z, \infty)$$ disjunkt, $U\cap V = \emptyset,\; U \neq \emptyset,\; V\neq \emptyset$, $U, V$ sind relativ-offen bzgl. $f(D)$ und $U\cap V = f(D)$ $$\implies f(D)\text{ nicht zusammenhängend \lightning.}$$ \end{proof} -\section{Vektor- und Matrix-wertige Funktionen} +\section{Vektor- und Matrixwertige Funktionen} \begin{align*} f:&D \subset \K^n \to \K^m\\ f:&D \subset \K^n \to \K^{r\times s}\\ f:&D \subset \K^{m\times n} \to \K^{r\times s} \end{align*} \begin{itemize} - \item Mit Hilfe von Normen in $\K^n$ und $K^{m\times n}$ sind die Definitionen von Stetigkeit etc. übertragbart auf Vektor- und Matrix-wertige Funktionen. $\implies$ solche stetigen Abbildungen auf kompakten Mengen sind gleichmäßig stetig und beschränkt. + \item Mit Hilfe von Normen in $\K^n$ und $\K^{m\times n}$ sind die Definitionen von Stetigkeit etc. übertragbart auf Vektor- und Matrixwertige Funktionen. $\implies$ solche stetigen Abbildungen auf kompakten Mengen sind gleichmäßig stetig und beschränkt. \item $f: D\subset \K^n \to \K^m$ stetig $\Leftrightarrow$ alle Komponenten $f_i:D\subset \K^n \to \K,\; i= 1, \dots, m$ sind stetig (genauso für $f: D\subset \K^{n\times m} \to \K^{r\times s}$ etc.) \end{itemize} \begin{lemma} Seien $g: D\subset \K^n\to B \subset \K^m$ und $f:B\to \K^r$ stetig. Dann ist die Komposition $f\circ g: D\subset \K^n \to \K^r$ stetig. \end{lemma} \begin{proof} - Sei $x\in D, x^{(k)} \in D$ s.d. $\lim\limits_{k\to\infty}x^{(k)} = x$. Es gilt + Sei $x\in D, \ x^{(k)} \subset D$ s.d. $\lim\limits_{k\to\infty}x^{(k)} = x$. Es gilt \begin{align*} g\text{ stetig }&\implies y^{(k)} = g(x^{(k)}) \underset{k\to \infty}{\to} g(x) =: y\\ f\text{ stetig }&\implies (f\circ g)\left(x^{(k)}\right) = f\left(g\left(x^{(k)}\right)\right) = f(y^{(k)}) \underset{k\to \infty}{\to} f(y) diff --git a/analysisII.pdf b/analysisII.pdf index 26b7449a421ff37628a219519d004ac121f9547e..0972e3dbd5ffa17754908dd1b8d1adb6a37f6897 100644 GIT binary patch delta 426419 zcmb@tWmsLywk;ZhhTs;Q2?X~!ad&rjcXuZPcMB4NhlJqnPH=a3CnN;7Bm{Uf$=Y|X zv)0=8op;{(HNIJl8r7;-tJbUa`Y4*Aj_RdWr&1P|WMX6HK&G0U?OQ}<2eE>jOl^_* z`9UmdAQmYnbte#uI7p8J+O7&>k+g9)00{~rTR58ke2eS6HIr;46Vt;E7c$wnez8J`^qn7cdMJh0z&6vQ~w zKlw~vtsD}zw$Xt$WI95i950_F>isB2N>%_#4`%?w(OIRm!M8oaa*$1i;_SjD@8j_soxxW}b=McVm`Uge?E-3PL2Hk()70=`Fac!~D+^>6F(*fN z3rBZ1(9g-LBD1JkxH);anpwDkIDa=NSeVl|}p8wnt z{Hr57D>FAI_;3Aw!u?}F5Oh}dPOfUsCT12O$WLG(C{A~G3s*4d(UIrafwxPGkufVn4aBk}f| zg?N>yc%rmxbpFQ9B!*RA#IdWrWd$tfffc+diC{5oEPfUeO zS<apSP9ZR)oWE&waWR|&GwBSmYIYpM-E{>dDk}^XBn}+*SG9xwdx>i 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