CRN 11378: HW 5
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'''Problem 23.''' Let $0<c<1$. 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 23.''' Let $0<c<1$. Assume $f:\mathbb{R}\to\mathbb{R}$ satisfies $|f(x)-f(y)|\leq c\cdot |x-y|$ for all $x,y\in\mathbb{R}$. | ||
| − | #Show that $f$ is continuous on $\mathbb{R}$. | + | #Show that $f$ is uniformly continuous on $\mathbb{R}$. |
#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 there is an $x\in\mathbb{R}$ such that $f(x)=x$. (Hint: for any $y\in\mathbb{R}$ look at the sequence $y,f(y),f(f(y)),f(f(f(y)))\ldots$. | ||
Revision as of 10:18, 5 November 2019
Problem 21. Let the function $f:\mathbb{R}\to\mathbb{R}$ be given by $f(x)=\sqrt[3]{x}$.
- Show that $f$ is continuous at $0$.
- Show that $f$ is continuous at any $x_0\neq 0$. (The identity $a^3-b^3=(a-b)(a^2+ab+b^2)$ will be helpful.)
Problem 22. Assume $f:\mathbb{R}\to\mathbb{R}$ is continuous on $\mathbb{R}$. Show that $\{x\in\mathbb{R}\ |\ f(x)=0\}$ is a closed set.
Problem 23. Let $0<c<1$. Assume $f:\mathbb{R}\to\mathbb{R}$ satisfies $|f(x)-f(y)|\leq c\cdot |x-y|$ for all $x,y\in\mathbb{R}$.
- Show that $f$ is uniformly continuous on $\mathbb{R}$.
- 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 $y,f(y),f(f(y)),f(f(f(y)))\ldots$.
Problem 24.
- 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 fails without the boundedness condition.
