From 44156909a9492d33e16c5c894c354dc4ac6702bd Mon Sep 17 00:00:00 2001 From: Robert Date: Thu, 25 Mar 2021 18:55:20 +0100 Subject: [PATCH] Finished section sets and functions --- chapters/sections/lims_and_funcs.tex | 576 +++++++++++++++++++++++++++ script.pdf | Bin 550880 -> 595301 bytes script.tex | 3 + 3 files changed, 579 insertions(+) diff --git a/chapters/sections/lims_and_funcs.tex b/chapters/sections/lims_and_funcs.tex index db4ceee..6b9406f 100644 --- a/chapters/sections/lims_and_funcs.tex +++ b/chapters/sections/lims_and_funcs.tex @@ -49,4 +49,580 @@ In this chapter we will introduce the notation x \in \realn \text{ is a boundary point of } D \subset \realn \iff \exists \anyseqdef{D} \text{ such that } x_n \rightarrow x \] \end{thm} +\begin{proof} + Let $x$ be a boundary point of $D$. Then + \begin{equation} + \forall n \in \natn ~\exists x_n \in D \cap \left( x - \frac{1}{n}, x+ \frac{1}{n} \right) + \end{equation} + The resulting sequence $\seq{x}$ is in $D$, and + \begin{equation} + |x - x_n| \le \frac{1}{n} + \end{equation} + holds. Therefore, $x_n$ converges to $x$. Now let $\anyseqdef{D}$, with $x_n \rightarrow x$. This means + \begin{equation} + \forall \epsilon > 0 ~\exists N \in \natn: ~~|x - x_N| < \epsilon + \end{equation} + Then + \begin{equation} + x_N \in D \cap \oball(x) + \end{equation} + Since $\epsilon$ is arbitrary, $x$ is a boundary point of $D$. +\end{proof} + +\begin{defi} + Let $D \subset \realn$ and $f: D \rightarrow \realn$. Let $x_0$ be a boundary point of $D$. + We say that $f$ converges to $y \in \realn$ for $x \rightarrow x_0$ and write + \[ + \lim_{x \rightarrow x_0} f(x) = y + \] + if + \[ + \forall \epsilon > 0 ~\exists \delta > 0: ~~|x-x_0| < \delta \implies |f(x) - f(y)| < \epsilon + \] +\end{defi} + +\begin{rem} + This definition only makes sense for boundary points $x_0$ of $D$. The most imoprtant case is + \[ + D = (x_0 - a, x_0 + a) \setminus \set{x_0} + \] +\end{rem} + +\begin{eg} +\begin{enumerate}[(i)] + \item Let $a \in \realn$ + \begin{align*} + f: \realn &\longrightarrow \realn \\ + x &\longmapsto ax + \end{align*} + Consider $a \ne 0$: Let $\epsilon > 0$. We want that + \[ + |f(x) - 0| = |a||x| \stackrel{!}{<} \epsilon + \] + Choose $\delta = \frac{\epsilon}{|a|}$. Then we have + \[ + |x| = |x - 0| < \delta \implies |f(x) - 0| = |a||x| < |a| \delta = |a| \frac{\epsilon}{|a|} = \epsilon + \] + Therefore + \[ + \lim_{x \rightarrow 0} f(x) = 0 + \] + + \item Consider + \begin{align*} + f: \realn &\longrightarrow \realn \\ + x &\longmapsto \begin{cases} + 1 ,& x > 0 \\ + -1 ,& x < 0 \\ + \end{cases} + \end{align*} + $f$ doesn't converge for $x \rightarrow 0$. Assume $y \in \realn$ is the limit of $x$ at $0$. This means that there is a $\delta > 0$ such that + \[ + |f(x) - y| < 1 \text{ if } |x| = |x - 0| < \delta + \] + Then, for any $x \in (0, \delta)$ we have + \[ + 2 = |f(x) - f(-x)| \le \underbrace{|f(x) - y|}_{< 1} + \underbrace{|y - f(-x)|}_{< 1} < 2 + \] + which is a contradiction. +\end{enumerate} +\end{eg} + +\begin{thm} + Let $f: D \rightarrow \realn$, $x_0$ a boundary point of $D$ and $y \in \realn$. Then + \[ + \lim_{x \rightarrow x_0} f(x) = y \iff \forall \anyseqdef{D} \text{ with } x_n \longrightarrow x_0: ~~\limn f(x_n) = x_0 + \] +\end{thm} +\begin{proof} + Assume that $\lim_{x \rightarrow x_0} f(x)$ and that there is $\anyseqdef{D}$ converging to $x$. + Let $\epsilon > 0$, then + \begin{equation} + \exists \delta > 0: ~~|x - x_0| < \delta \implies |f(x) - y| < \epsilon + \end{equation} + Since $x_n \rightarrow x_0$, we know that + \begin{equation} + \exists N \in \natn ~\forall n > N: ~~|x_n - x_0| < \delta + \end{equation} + For such $n$, the epsilon criterion $|f(x_n) - y| < \epsilon$ also holds, and thus + \begin{equation} + f(x_n) \convinf y + \end{equation} + Now to prove the "$\impliedby$" direction, assume that $\lim_{x \rightarrow x_0} f(x) \ne y$, i.e. + \begin{equation} + \exists \epsilon > 0 ~\forall \delta > 0 ~\exists x \in D: ~~|x - x_0| < \delta \wedge |f(x) - y| \ge \epsilon + \end{equation} + Choose $\forall x \in \natn$ one $x_n$ such that + \begin{equation} + |x_n - x_0| < \frac{1}{n} \text{ but } |f(x_n) - y| \ge \epsilon + \end{equation} + Then $x_n \rightarrow x_0$, but $|f(x_n) - y| \ge \epsilon ~\forall n \in \natn$, so + \begin{equation} + \limn f(x_n) \ne y + \end{equation} + This indirectly proves "$\impliedby$". +\end{proof} + +\begin{eg} + Consider $D = \realn \subset \set{0}$, we want to prove + \[ + \lim_{x \rightarrow 0} \frac{1}{1-x} = 1 + \] + So let $\anyseqdef{D}$ with $x_n \rightarrow 0$. Then + \[ + \frac{1}{1-x_n} \convinf 1 + \] + \[ + \implies \lim_{x \rightarrow 0} \frac{1}{1-x} = 1 + \] + However, the limit $\lim_{x \rightarrow 1}$ doesn't exist. Let $x_n = \frac{1}{n} + 1$ with $x_n \rightarrow 1$. Then + \[ + \frac{1}{1 - (\frac{1}{n} + 1)} = -n \convinf -\infty + \] + This doesn't converge, thus there is no limit. +\end{eg} + +\begin{cor} + Let $f, g: D \rightarrow \realn$, $x_0$ a boundary point and $y, z \in \realn$ such that + \begin{align*} + \lim_{x \rightarrow x_0} f(x) = y && \lim_{x \rightarrow x_0} g(x) = z + \end{align*} + Then + \begin{align*} + \lim_{x \rightarrow x_0} (f(x) + g(x)) &= y+z \\ + \lim_{x \rightarrow x_0} (f(x) \cdot g(x)) &= y \cdot z + \end{align*} + If $z \ne 0$, then + \[ + \lim_{x \rightarrow x_0} (\frac{f(x)}{g(x)}) = \frac{y}{z} + \] +\end{cor} +\begin{proof} + Here we will only prove the last statement. + Let $\lim_{x \rightarrow x_0} = z \ne 0$. Then + \begin{equation} + \exists \delta > 0 ~\forall x \in \oball[\delta](x_0): ~~|g(x) - z| < |z| + \end{equation} + $g$ doesn't have any roots on $\oball[\delta](x_0)$. Let $\anyseqdef{D \cap \oball[\delta](x_0)}$ converge to $x_0$. + According to prerequisites, we have + \noindent\begin{subequations} + \begin{multicols}{2} + \noindent + \begin{equation} + \limn f(x_n) = y + \end{equation} + \begin{equation} + \limn g(x_n) = z \ne 0 + \end{equation} + \end{multicols} + \end{subequations} + \noindent Thus + \begin{equation} + \implies \limn \frac{f(x_n)}{g(x_n)} = \frac{y}{z} \implies \lim_{x \rightarrow x_0} \frac{f(x)}{g(x)} = \frac{y}{z} + \end{equation} +\end{proof} + +\begin{cor}[Squeeze Theorem] + Let $f, g, h: D \rightarrow \realn$ and $x$ a boundary point of $D$. If for $y \in \realn$ + \[ + \lim_{x \rightarrow x_0} f(x) = y = \lim_{x \rightarrow x_0} h(x) + \] + and + \[ + f(x) \le g(x) \le h(x) ~~\forall x \in \oball(x_0) + \] + then + \[ + \lim_{x \rightarrow x_0} g(x) = y + \] +\end{cor} + +\begin{eg} + Consider $\exp(x)$. We already know that + \[ + 1 + x \le \exp(x) ~~\forall x \in \realn + \] + This also implies that + \[ + 1 - x \le \exp(-x) = \frac{1}{\exp(x)} ~~\forall x \in \realn + \] + So + \[ + 1 + x \le \exp(x) \le \frac{1}{1 - x} + \] + The limits of these terms are + \begin{align*} + \lim_{x \rightarrow 0} (1+x) = 1 && \lim_{x \rightarrow 0} \left(\frac{1}{1-x}\right) = 1 + \end{align*} + And using the squeeze theorem this results in + \[ + \lim_{x \rightarrow 0} \exp(0) = 1 + \] +\end{eg} + +\begin{defi} + Let $f: D \rightarrow \realn$ and $x_0$ a boundary point of $D$. We say $f$ diverges to infinity for $x \rightarrow x_0$ and write + \[ + \lim_{x \rightarrow x_0} f(x) = \infty + \] + if + \[ + \forall K \in (0, \infty) ~\exists \delta > 0: ~~|x - x_0| < \delta \implies f(x) \ge K + \] +\end{defi} + +\begin{defi} + Let $D \subset \realn$ be unbounded above. We say $f$ converges for $x \rightarrow \infty$ to $y \in \realn$ and write + \[ + \lim_{x \rightarrow \infty} f(x) = y + \] + if + \[ + \forall \epsilon > 0 ~\exists K \in (0, \infty) ~\forall x > K: ~~|f(x) - y| < \epsilon + \] +\end{defi} + +\begin{rem} + Let $f: D \rightarrow \cmpln$ and $x_0$ a boundary point of $D$. Then + \begin{align*} + &\limes{x}{x_0} f(x) = y \in \cmpln \\ + \iff & \limes{x}{x_0} \Re(f(x)) = \Re(y) \wedge \limes{x}{x_0} \Im(f(x)) = \Im(y) \\ + \iff & \limes{x}{x_0} |f(x) - y| = 0 + \end{align*} +\end{rem} + +\begin{defi} + Let $D \subset K$, $f:D \rightarrow K$ and $x_0 \in D$. $f$ is called continuous in $x_0$ if + \[ + \forall \epsilon > 0 ~\exists \delta > 0: ~~|x - x_0| < \delta \implies |f(x) - f(x_0)| < \epsilon + \] + If $f$ is continuous in every point of $D$, we call $f$ continuous. + + $f$ is called Lipschitz + continuous if + \[ + \exists L \in (0, \infty) ~\forall x, y \in D: ~~|f(x) - f(y)| \le L|x - y| + \] + $L$ is called Lipschitz constant +\end{defi} + +\begin{rem} + Let $f: D \rightarrow \field$ + \[ + f \text{ is continuous in } x_0 \in D \iff \limes{x}{x_0} f(x) = f(x_0) + \] +\end{rem} + +\begin{eg} + We want to show that + \begin{align*} + f: \realn &\longrightarrow \realn \\ + x &\longmapsto x^2 + \end{align*} + is continuous. To do that, let $x_0 \in \realn$, $\epsilon > 0$. We want + \[ + |f(x) - f(x_0)| = |x^2 - x_0^2| = |x - x_0||x + x_0| \must[\le] \epsilon + \] + So we choose + \[ + \delta = \min \set{1, \frac{\epsilon}{2|x_0| + 1}} > 0 + \] + Then for every $x$ with $|x - x_0| < \delta$ + \begin{align*} + |f(x) - f(x_0)| &= |x - x_0||x + x_0| \le \delta(|x| + |x_0|) \le \delta(|x_0| + \delta + |x_0|) \\ + &\le \delta(2|x_0| + 1) \le \frac{\epsilon}{2|x_0| + 1}(2|x_0| + 1) = \epsilon + \end{align*} +\end{eg} + +\begin{thm} + Every Lipschitz continuous function is continuous +\end{thm} +\begin{proof} + Let $f: D \rightarrow \field$ be a Lipschitz continuous function with Lipschitz constant $L > 0$. I.e. + \begin{equation} + \forall x, y \in D: ~~|f(x) - f(y)| \le L|x - y| + \end{equation} + Let $x_0 \in \realn$ and $\epsilon > 0$. Choose $\delta = \frac{\epsilon}{L}$. Then $|x - x_0| < \delta$ implies + \begin{equation} + |f(x) - f(x_0)| \le L|x - x_0| \le L \cdot \delta = \epsilon + \end{equation} +\end{proof} + +\begin{eg} + \begin{enumerate}[(i)] + \item Consider the constant function $x \mapsto c$, $c \in \field$. + \[ + |f(x) - f(y)| = |c - c| = 0 \le 1 \cdot |x - y| + \] + + \item Consider the linear function $x \mapsto cx$, $c \in \field$. + \[ + |f(x) - f(y)| = |cx - cy| = |c||x - y| + \] + \end{enumerate} + These two functions are Lipschitz continuous, and therefore continuous. + \begin{enumerate}[(i)]\setcounter{enumi}{2} + \item Consider $x \mapsto \Re(x)$. Then + \[ + |\Re(x) - \Re(y)| = |\Re(x - y)| \le |x - y| + \] + Analogously this works for $\Im(x)$. Both of those are Lipschitz continuous. + + \item Lipschitz continuity depends on $D$. E.g. + \begin{align*} + f: [0, 1] &\longrightarrow \realn \\ + x &\longmapsto x^2 + \end{align*} + is Lipschitz continuous: + \[ + |f(x) - f(y)| = |x-y||x+y| \le 2 \cdot |x - y| + \] + However, + \begin{align*} + g: \realn &\longrightarrow \realn \\ + x &\longmapsto x^2 + \end{align*} + is NOT Lipschitz continuous, because + \[ + \frac{|g(n+1) - g(n)|}{(n+1) - n} = 2n + 1 \convinf \infty + \] + \end{enumerate} +\end{eg} + +\begin{rem} + Let $f: D \rightarrow \field$. + \begin{gather*} + f \text{ is continuous in } x_0 \in D \\ + \iff \\ + \forall \anyseqdef{D} \text{ with } x_n \rightarrow x_0: ~~\limn f(x_n) = f(x_0) + \end{gather*} + If $f, g$ are continuous in $x_0$, then $f+g$ and $f \cdot g$ are also continuous in $x_0$, and if $g(x_0) \ne 0$ then $f/g$ is also continuous in $x_0$. + Notably, polynomials are continuous. A rational function (the quotient of two polynomials) is continuous in all points that are not roots of the denominator. +\end{rem} + +\begin{thm} + Let $D \subset \field$, and let + \begin{subequations} + \begin{gather} + f: D \longrightarrow \field \text{ continuous in } x_0 \in D \\ + g: f(D) \longrightarrow \field \text{ continuous in } f(x_0) + \end{gather} + \end{subequations} + Then $g \circ f$ is also continuous in $x_0$. +\end{thm} +\begin{proof} + Let $\epsilon > 0$. Since $g$ is continuous in $f(x_0)$, + \begin{equation} + \exists \delta_1 > 0: ~~|y - f(x_0)| < \delta_1 \implies |g(y) - g(f(x_0))| < \epsilon + \end{equation} + Since $f$ is continuous in $x_0$, + \begin{equation} + \exists \delta_2 > 0: ~~|x - x_0| < \delta_2 \implies |f(x) - f(x_0)| < \delta_1 + \end{equation} + For such $x$ the following holds + \begin{equation} + |(g \circ f)(x) - (g \circ f)(x_0)| = |g(f(x)) - g(f(x_0))| < \epsilon + \end{equation} + which implies that $g \circ f$ is continuous in $x_0$. +\end{proof} + +\begin{eg} + Consider the following mappings + \begin{align*} + &f: \realn \longrightarrow \realn, ~~x \longmapsto |x| \\ + &g: \realn \longrightarrow \realn \setminus \set{-1}, ~~y \longmapsto \frac{1 - y}{1 + y} \\ + &h: \realn \longrightarrow \realn, ~~x \longmapsto \frac{1 - |x|}{1 + |x|} + \end{align*} + It is clear that $h = g \circ f$. Since $f$, $g$ are continuous, $h$ must also be continuous. +\end{eg} + +\begin{eg} + The functions $\exp$, $\sin$ and $\cos$ are continuous. We know that + \[ + \limes{h}{0} \frac{\exp(k) - 1}{h} = 1 + \] + From this follows that + \[ + \limes{h}{0} \exp(k) = \exp(0) = 0 + \] + Thus, $\exp$ is continuous in $0$. Let $x_0 \in \realn$, then + \begin{align*} + \limes{x}{x_0} \exp(x) &= \limes{h}{0} \exp(x_0 + h) = \limes{h}{0} \exp(x_0)\exp(h) \\ + &= \exp(x_0) - \limes{h}{0} \exp(h) = \exp{x_0} + \end{align*} + Now, consider the function $x \mapsto \exp(ix)$. For $x_0 \in \realn$ + \begin{align*} + |\underbrace{\exp(i(x_0 + h))}_{\exp(ix_0) \dot \exp(ih)} - \exp(i h_0)| &= \underbrace{|\exp(ix_0)|}_1|\exp(ih) - 1| \\ + &\le 1 \cdot \left| \sum_{k = 0}^{\infty} \frac{(ih)^k}{k!} - 1 \right| = \left| \series{k} \frac{(ih)^k}{k!} \right| \\ + &\le \series{k} \left| \frac{(ih)^k}{k!} \right| \\ + &= \series{k} \frac{|h|^k}{k!} = \sum_{k = 0}^{\infty} \frac{|h|^k}{k!} - 1 = \exp(|h|) - 1 + \end{align*} + For $h \rightarrow 0$, the absolute function converges $|h| \rightarrow 0$, and therefore + \[ + \lim{h}{0} |\exp(i(x_0 + h)) - \exp(ix)| = 0 + \] + due to the squeeze theorem. I.e., $x \mapsto \exp(ix)$ is also continuous. Thus + \begin{align*} + \cos x = \Re(\exp(ix)) && \sin x = \Im(\exp(ix)) + \end{align*} + are also continuous due to the concatination of continuous functions. +\end{eg} + +\begin{lem} + Let $a, b \in \realn$ with $a < b$, and let + \[ + f: [a, b] \longrightarrow \realn + \] + be a continuous function. Furthermore, let $y \in \realn$. Now if the set + \[ + \set[f(x) \ge y]{x \in [a, b]} + \] + is non-empty, it has a smallest element. +\end{lem} +\begin{proof} + Let $M$ be non-empty. Set $x_0 = \inf\set{M}$. Then it is to be shown that $x_0 \in M$, or that $f(x_0) \ge y$. + There exists a sequence $\anyseqdef{M}$ such that $x_n \rightarrow x_0$. Because of the continuity of $f$, + \begin{equation} + f(x_0) = f(\limn x_n) = \limn f(x_n) \ge y + \end{equation} + holds, thus $x_0 \in M$. +\end{proof} + +\begin{thm}[Extreme value theorem] + Let $a, b \in \realn$ with $a < b$, and let $f: [a, b] \rightarrow \realn$ continuous. + Then the function $f$ attains a maximum, i.e. + \[ + \exists x_0 \in [a, b] ~\forall x \in [a, b]: ~~f(x) \le f(x_0) + \] +\end{thm} +\begin{proof} + First we show that $f$ is bounded. Assume $f$ is unbounded above, i.e. + \begin{equation} + \set[f(x) \ge n]{x \in [a, b]} =: M_n, ~~n \in \natn + \end{equation} + According to the last lemma, every $M_n$ has a smallest element $x_n$. + The sequence $(x_n)_{n \in \natn}$ is monotonically increasing ($M_{n+1} \subset M_n$) and bounded above by $b$. + Thus, $x_n$ converges to some $x_0 \in [a, b]$. Now consider the sequence $(f(x_n))_{n \in \natn}$. By definition + \begin{equation} + \limn f(x_n) \ge \limn n = \infty + \end{equation} + And since $f$ is continuous, $\limn f(x_n) = f(x_0)$ must hold. This contradicts the assumption, so $f$ is bounded. + + Now set + \begin{equation} + y = \sup\set[{x \in [a, b]}]{f(x)} + \end{equation} + In case $f$ is equal to $y$ everywhere, there is nothing to show. So assume that there are values for which $f \ne y$. + According to the definition of the supremum, the sets + \begin{equation} + \set[f(x) \ge y - \frac{1}{n}]{x \in [a, b]} + \end{equation} + are non-empty for all $n \in \natn$, and thus they have a smallest element $x_n$. + The sequence $(x_n)_{n \in \natn}$ is monotonically increasing and bounded, i.e. it converges to $x_0 \in [a, b]$. + Therefore + \begin{equation} + y \ge f(x_0) = \limn f(x_n) \ge \limn y - \frac{1}{n} = y + \end{equation} + From this follows + \begin{equation} + f(x_0) = y \implies f(x_0) \text{ upper bound of } \set[{x \in [a, b]}]{f(x)} + \end{equation} +\end{proof} + +\begin{thm}[Intermediate value theorem] + Let $a, b \in \realn$ with $a < b$, and $f: [a, b] \rightarrow \realn$ a continuous function with $f(a) < f(b)$. + \[ + y \in (f(a), f(b)) \implies \exists x_0 \in (a, b): ~~f(x_0) = y + \] +\end{thm} +\begin{proof} + \noproof +\end{proof} + +\begin{eg} + $\cos$ has in $[0, 2]$ exactly one root. Consider the definition + \[ + \cos x = \sum_{k = 0}^{\infty} \frac{(-1)^k x^{2k}}{(2k)!} + \] + We know that $\cos 0 = 1$. Furthermore we can show that + \[ + -1 = \underbrace{1 - \frac{2^2}{2!}}_{\text{2nd partial sum}} \le \cos(2) \le \underbrace{1 - \frac{2^2}{2!} + \frac{2^4}{4!}}_{\text{3rd partial sum}} < 0 + \] + The intermediate value theorem tells us that there exists a root in $[0, 2]$. Now we need to show that $\cos$ is strictly monotonically decreasing on $[0, 2]$. + Choose $z \in [0, 2]$. Then + \[ + z \le \sin z \le z - \frac{z^3}{3!} + \] + The addition theorem tells us that + \[ + \cos(x) - \cos(y) = -2 \sin\left(\frac{x+y}{2}\right) \sin\left(\frac{x-y}{2}\right) < 0 + \] + for $x, y \in (0, 2]$ and $x > y$. Thus $\cos$ is strictly monotonically decreasing on $[0, 2]$. +\end{eg} + +\begin{cor} + Let $I$ be an interval and $f: I \rightarrow \realn$ continuous. Then $f(I)$ is also an interval. +\end{cor} +\begin{proof} + \reader +\end{proof} + +\begin{thm} + Let $I$ be an interval, $f: I \rightarrow \realn$ continuous. If $f$ is strictly monotonically increasing, then the inverse function + $\inv{f}: f(I) \rightarrow I$ exists and is continuous. +\end{thm} +\begin{hproof} + $f(I)$ is an interval, and $f$ is injective. This is because if $f(x) = f(\tilde{x})$, then $x = \tilde{x}$ or else $f$ wouldn't be strictly monotonic. + This means + \begin{equation} + \exists g: f(I) \longrightarrow \realn: ~~f(x) = y \iff g(y) = x + \end{equation} + Let $y_0 \in f(I)$ and $\epsilon > 0$. We require that $x_0$ is not a boundary point of $I$. Then choose $0 < \tilde{\epsilon} < \epsilon$ such that + $(x_0 - \tilde{\epsilon}, x_0 + \tilde{epsilon}) \in I$. Choose + \begin{equation} + \delta = \min \set{\underbrace{f(x_0 + \tilde{\epsilon}) - y_0}_{> 0}, \underbrace{y_0 - f(x_0 - \tilde{\epsilon})}_{> 0}} > 0 + \end{equation} + If $y \in f(I)$ with $|y - y_0| < \delta$ then + \begin{equation} + f(x_o - \tilde{epsilon}) \le x_0 - \delta < y < y_0 + \delta \le f(x_0 + \tilde{\epsilon}) + \end{equation} + From the strict monotony of $g$ we can conclude + \begin{equation} + x_0 - \tilde{epsilon} < g(y) < x_0 + \tilde{\epsilon} + \end{equation} + so + \begin{equation} + |g(y) - g(y_0)| = |g(y) - x_0| < \tilde{\epsilon} < \epsilon + \end{equation} + Thus, $g$ is continuous in $y_0$. Since $y_0 \in f(I)$ was chose arbitrarily, all of $g$ is continuous. + To prove the monotony of $g$, assume $y < \tilde{y}$ and $g(y) \ge g(\tilde{y})$ for $y, \tilde{y} \in f(I)$. From the monotony of $f$ we know that + \begin{equation} + y \ge \tilde{y} + \end{equation} + which is a contradiction, so $g$ is strictly monotonic. +\end{hproof} + +\begin{eg} + \begin{enumerate}[(i)] + \item Let $n \in \natn$ and consider + \begin{align*} + f: [0, \infty) &\longrightarrow \realn \\ + x &\longmapsto x^n + \end{align*} + $f$ is continuous and strictly monotonically increasing. Thus the inverse function + \[ + \sqrt[n]{\cdot}: [0, \infty) \longrightarrow \realn^+ + \] + is also continuous. + + \item Consider $\exp: \realn \rightarrow \realn$. It's clear that $\exp(\realn) = (0, \infty)$, so the mapping + \[ + \ln: (0, \infty) \rightarrow \realn + \] + is continuous and strictly monotonically increasing. + + \item Equal arguments can be made for the trigonometric functions. + \end{enumerate} +\end{eg} + \end{document} \ No newline at end of file diff --git a/script.pdf b/script.pdf index 2b87ea7a42a1e627b764801b4718688b3eea4cff..936cb0986cf2f762209a0bf2ca15159e325e3a16 100644 GIT binary patch delta 120932 zcmaHzbx>T}*6ne3*Wj)VG!oq1-Q8V7uqJ4NYZ`ag;7-us5G(`;?(PACy-v=(@7;6H z`#%2YV)mMI@6}Wn^{X+)+UrwPI~BCSVbEbPjj+38;9)d4H1LxT3{s080-PH{^0XK= zA*TS4l9N&cg+s&I)79I<(%O>}z$w5DhDL{Dr04mwBqD-h?PBF>;eY~J;f1{Q`&S_$ z6b>m$L&!Tk$rnS+;J4^-xXI#@xZrbUSR!!CzpE8=I8v|>IvmcQHF69{b@%Ve7z2(N z%!>hs_h&5|1C9jji}A0{AHO?WepiXXcwll&$lC3{E92i4ALgG>mF-T$=@xDCkpk%@chDubJQ{gC4de51mTyrZlBqF{byF$+v{H9mXb)Nb9cV$a|oIG!G*? z>xN$0Tc{e%>DP4bk$&*5I!MGwlE`Y`+3u_LsYCmTXqL`blI{_#Q%bUcU`*VRvH7h_ zhvw(GA8*s-bZU0ToUhIf+woa{B$H6bt~&eW;N-Z&bX}h@7U7#oaS=0NIl~P^1yRT| 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