CRN 11247: HW 6 - Revision history

HelmutKnaust at 04:40, 9 November 2021

2021-11-09T04:40:31Z

HelmutKnaust at 04:39, 9 November 2021

2021-11-09T04:39:40Z

HelmutKnaust: Created page with "'''Problem 26.''' Let $f,g:\mathbb{R}\to\mathbb{R}$ be two continuous functions. Define $h(x)=\max\{f(x),g(x)\}$ for all $x\in\mathbb{R}$. Show that $h$ is continuous on $\ma..."

2021-11-09T04:30:02Z

Created page with "'''Problem 26.''' Let $f,g:\mathbb{R}\to\mathbb{R}$ be two continuous functions. Define $h(x)=\max\{f(x),g(x)\}$ for all $x\in\mathbb{R}$. Show that $h$ is continuous on $\ma..."

New page

'''Problem 26.'''
Let $f,g:\mathbb{R}\to\mathbb{R}$ be two continuous functions. Define $h(x)=\max\{f(x),g(x)\}$ for all $x\in\mathbb{R}$. Show that $h$ is continuous on $\mathbb{R}$.

'''Problem 27.'''
Let $f:\mathbb{R}\to\mathbb{R}$ be continuous on $\mathbb{R}$, and assume that for all $\varepsilon>0$ there is an $N>0$ such that $|f(x)|<\varepsilon$ for all $x$ satisfying $|x|>N$. Show that $f$ is uniformly continuous on $\mathbb{R}$.

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 Let $f:\mathbb{R}\to\mathbb{R}$ be continuous on $\mathbb{R}$, and assume that for all $\varepsilon>0$ there is an $N>0$ such that $|f(x)|<\varepsilon$ for all $x$ satisfying $|x|>N$. Show that $f$ is uniformly continuous on $\mathbb{R}$.
-'''Problem 27.''' Show: If a function $f:[a,b]\to\mathbb{R}$ is increasing and satisfies the intermediate value property, then $f$ is continuous on $[a,b]$. (Ex. 4.5.3)
+'''Problem 27.''' Show: If a function $f:[a,b]\to\mathbb{R}$ is increasing and satisfies the intermediate value property, then $f$ is continuous on $[a,b]$. (Ex. 4.5.3.)
-'''Problem 28.''' Show: Let the function $f:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$ and let $F=\{x\in [a,b]\ |\mbox{ there is a } y\neq x\in [a,b] \mbox{ with } f(x)=f(y)\}$. Show that $F$ is empty or uncountable. (Ex. 4.5.4.)
+'''Problem 28.''' Let the function $f:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$ and let $F=\{x\in [a,b]\ |\mbox{ there is a } y\neq x\in [a,b] \mbox{ with } f(x)=f(y)\}$. Show that $F$ is empty or uncountable. (Ex. 4.5.4.)

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-'''Problem 26.'''
+'''Problem 25.'''
 Let $f,g:\mathbb{R}\to\mathbb{R}$ be two continuous functions. Define $h(x)=\max\{f(x),g(x)\}$ for all $x\in\mathbb{R}$. Show that $h$ is continuous on $\mathbb{R}$.
-'''Problem 27.'''
+'''Problem 26.'''
 Let $f:\mathbb{R}\to\mathbb{R}$ be continuous on $\mathbb{R}$, and assume that for all $\varepsilon>0$ there is an $N>0$ such that $|f(x)|<\varepsilon$ for all $x$ satisfying $|x|>N$. Show that $f$ is uniformly continuous on $\mathbb{R}$.