CRN 11247: HW 6
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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 $\mathbb{R}$. | 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.''' | + | '''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.) |
Latest revision as of 22:40, 8 November 2021
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 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 $\mathbb{R}$.
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. 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.)
