CRN 11378: HW 4
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| − | '''Problem | + | '''Problem 25.''' |
| − | + | Let $f:[a,b]\to\mathbb{R}$ be an increasing function. Show that $\lim_{x\to a}f(x)$ exists. What can you say about the relationship between this limit and $f(a)$? | |
| − | + | ||
| − | '''Problem | + | '''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 | + | '''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 $\mathbb{R}$. |
| − | + | ||
| − | '''Problem | + | '''Problem 28.''' |
| − | + | Let $f:[a,b]\to\mathbb{R}$ be a function. We say $f$ satisfies $(*)$ if there is an $M>0$ such that $|f(x)-f(y)|\leq M\cdot |x-y|$ for all $x,y\in [a,b]$. | |
| − | + | # Let $g:[0,1]\to\mathbb{R}$ be given by $g(x)=\sqrt{x}$. Show that $g$ does not satisfy $(*)$. | |
| − | + | # Is $g$ uniformly continuous on $[a,b]$? Is $g$ uniformly continuous on $(a,b)$? Explain! | |
| − | + | ||
Revision as of 12:12, 12 November 2019
Problem 25. Let $f:[a,b]\to\mathbb{R}$ be an increasing function. Show that $\lim_{x\to a}f(x)$ exists. What can you say about the relationship between this limit and $f(a)$?
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 $\mathbb{R}$.
Problem 28. Let $f:[a,b]\to\mathbb{R}$ be a function. We say $f$ satisfies $(*)$ if there is an $M>0$ such that $|f(x)-f(y)|\leq M\cdot |x-y|$ for all $x,y\in [a,b]$.
- Let $g:[0,1]\to\mathbb{R}$ be given by $g(x)=\sqrt{x}$. Show that $g$ does not satisfy $(*)$.
- Is $g$ uniformly continuous on $[a,b]$? Is $g$ uniformly continuous on $(a,b)$? Explain!
