CRN 11378: HW 5

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'''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 $x,y\in\mathbb{R}$.
 
'''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 $\mathbb{R}$.
#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 $y,f(y),f(f(y)),f(f(f(y)))\ldots$.
+
#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 $c=1$.
 
#Show that the result in 2. above may fail if $c=1$.
  

Latest revision as of 15:55, 5 November 2019

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. 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 $x\in\mathbb{R}$.

  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.
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