CRN 11247: HW 5

Problem 21. Let the function $f:\mathbb{R}\to\mathbb{R}$ be given by $f(x)=\sqrt[3]{x}$.

1. Show that $f$ is continuous at $0$.
2. 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 $x,y\in\mathbb{R}$.

1. Show that $f$ is uniformly continuous on $\mathbb{R}$.
2. 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 $y,f(y),f(f(y)),f(f(f(y))),\ldots$.
3. 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 $x\in D$.

1. 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 $\mathbb{R}$.
2. Show that the result may fail without the boundedness condition.
3. 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 $[0,1]$.