CRN 11378: HW 5 - Revision history

HelmutKnaust at 15:47, 28 October 2021

2021-10-28T15:47:09Z

2021-10-28T15:46:35Z

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2021-10-28T15:41:47Z

2019-11-05T21:55:05Z

2019-11-05T16:29:44Z

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 '''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 $\matbb{R}$. 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.
 #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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 '''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 $f,g:(0,1)\to\mathbb{R}$ be two bounded functions that are uniformly continuous on $(0,1)$. Show that $f\cdot g$ is uniformly continuous on $(0,1)$.
+#Let $f,g:\mathbb{R}\to\mathbb{R}$ be two bounded functions that are uniformly continuous on $\matbb{R}$. 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 $[0,1]$.

@@ Line 10: / Line 10: @@
 #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 $x\in D$.
+'''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 $f,g:(0,1)\to\mathbb{R}$ be two bounded functions that are uniformly continuous on $(0,1)$. Show that $f\cdot g$ is uniformly continuous on $(0,1)$.
 #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 $[0,1]$.

@@ Line 10: / Line 10: @@
 #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.

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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 $x,y\in\mathbb{R}$.
 #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 $c=1$.

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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 23.''' Let $0<c<1$. Assume $f:\mathbb{R}\to\mathbb{R}$ satisfies $|f(x)-f(y)|\leq c\cdot |x-y|$ for all $x,y\in\mathbb{R}$.
+'''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 $f$ 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 $y,f(y),f(f(y)),f(f(f(y)))\ldots$.
-#Show that the result above may fail if $c=1$.
+#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 fails without the boundedness condition.