CRN 11378: HW 5
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'''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 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 $ | + | '''Problem 23.''' Let $c\geq 0$. Assume $f:\mathbb{R}\to\mathbb{R}$ satisfies $|f(x)-f(y)|\leq c\cdot |x-y|$ for all $x,y\in\mathbb{R}$. |
#Show that $f$ is uniformly 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 $y,f(y),f(f(y)),f(f(f(y)))\ldots$. | + | #Now 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 24.''' | + | '''Problem 24.''' 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 $\mathbb{R}$. | #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 | + | #Show that the result may fail without the boundedness condition. |
| + | #Let $f,g:[0,1]\to\mathbb{R}$ be two functions that are uniformly continuous on $[0,1]$. Show that $f\cdot g$ is uniformly continuous on $[0,1]$. | ||
Latest revision as of 09:47, 28 October 2021
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 $c\geq 0$. Assume $f:\mathbb{R}\to\mathbb{R}$ satisfies $|f(x)-f(y)|\leq c\cdot |x-y|$ for all $x,y\in\mathbb{R}$.
- Show that $f$ is uniformly continuous on $\mathbb{R}$.
- Now 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 24. 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.
- Let $f,g:[0,1]\to\mathbb{R}$ be two functions that are uniformly continuous on $[0,1]$. Show that $f\cdot g$ is uniformly continuous on $[0,1]$.
