# CRN 11378: HW 5

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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 $0<c<1$. Assume $f:\mathbb{R}\to\mathbb{R}$ satisfies $|f(x)-f(y)|\leq c\cdot |x-y|$ for all $x,y\in\mathbb{R}$.

1. Show that $f$ is uniformly continuous on $\mathbb{R}$.
2. 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. A function $f:\mathbb{R} \to mathbb{R}$ is called bounded if there is an $M>0$ such that |f(x)|\leq M$for all$x\in\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}\$.

1. Show that the result fails without the boundedness condition.