CRN 10459: HW 6
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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 $\mathbb{R}$. | 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 $c\geq 0$. Assume $f:\mathbb{R}\to\mathbb{R}$ satisfies $|f(x)-f(y)|\leq c\cdot |x-y|$ for all $x,y\in\mathbb{R}$. | + | '''Problem 27.''' Let $c\geq 0$. Assume $f:\mathbb{R}\to\mathbb{R}$ satisfies $|f(x)-f(y)|\leq c\cdot |x-y|$ for all $x,y\in\mathbb{R}$. Note that such a function is uniformly continuous on $\mathbb{R}$. |
| − | + | #Assume that $0<c<1$. Show that there is an $x\in\mathbb{R}$ such that $f(x)=x$. (Hint: for any $y\in\mathbb{R}$ look at the sequence $y,f(y),f(f(y)),f(f(f(y))),\ldots$. | |
| − | # | + | |
#Show that the result in 2. above may fail if $c=1$. | #Show that the result in 2. above may fail if $c=1$. | ||
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'''Problem 29.''' | '''Problem 29.''' | ||
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 $\mathbb{R}$. | 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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'''Problem 30.''' 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 30.''' 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.) | ||
Latest revision as of 13:21, 11 November 2025
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 $c\geq 0$. Assume $f:\mathbb{R}\to\mathbb{R}$ satisfies $|f(x)-f(y)|\leq c\cdot |x-y|$ for all $x,y\in\mathbb{R}$. Note that such a function is uniformly continuous on $\mathbb{R}$.
- Assume that $0<c<1$. Show that there is an $x\in\mathbb{R}$ such that $f(x)=x$. (Hint: for any $y\in\mathbb{R}$ look at the sequence $y,f(y),f(f(y)),f(f(f(y))),\ldots$.
- Show that the result in 2. above may fail if $c=1$.
Problem 28. Let $D\subseteq\mathbb{R}$. A function $f:D \to \mathbb{R}$ is called bounded if there is an $M>0$ such that $|f(x)|\leq M$ for all $x\in D$.
- Let $f,g:\mathbb{R}\to\mathbb{R}$ be two bounded functions that are uniformly continuous on $\mathbb{R}$. Show that $f\cdot g$ is uniformly continuous on $\mathbb{R}$.
- Show that the result may fail without the boundedness condition.
Problem 29. 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 30. 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.)
