CRN 11247: HW 5 - Revision history

HelmutKnaust at 16:52, 4 November 2021

2021-11-04T16:52:37Z

HelmutKnaust at 15:51, 28 October 2021

2021-10-28T15:51:40Z

HelmutKnaust: Created page with "'''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\..."

2021-10-28T15:34:28Z

Created page with "'''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\..."

New page

'''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.''' 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 $\mathbb{R}$.
#Show that the result may fail without the boundedness condition.

← Older revision		Revision as of 16:52, 4 November 2021
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	#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 may fail without the boundedness condition.		#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 $[0,1]$.~~

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 #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 $x\in\mathbb{R}$.
+'''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.