http://helmut.knaust.info/mediawiki/index.php?title=CRN_11378:_HW_5&feed=atom&action=historyCRN 11378: HW 5 - Revision history2024-03-28T14:47:08ZRevision history for this page on the wikiMediaWiki 1.19.1http://helmut.knaust.info/mediawiki/index.php?title=CRN_11378:_HW_5&diff=3936&oldid=prevHelmutKnaust at 15:47, 28 October 20212021-10-28T15:47:09Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''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 $x\in D$.  </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''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 $x\in D$.  </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>#Let $f,g:\mathbb{R}\to\mathbb{R}$ be two bounded functions that are uniformly continuous on $\<del class="diffchange diffchange-inline">matbb</del>{R}$. Show that $f\cdot g$ is uniformly continuous on $\mathbb{R}$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>#Let $f,g:\mathbb{R}\to\mathbb{R}$ be two bounded functions that are uniformly continuous on $\<ins class="diffchange diffchange-inline">mathbb</ins>{R}$. Show that $f\cdot g$ is uniformly continuous on $\mathbb{R}$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that the result may fail without the boundedness condition.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that the result may fail without the boundedness condition.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#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 $[0,1]$.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#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 $[0,1]$.</div></td></tr>
</table>HelmutKnausthttp://helmut.knaust.info/mediawiki/index.php?title=CRN_11378:_HW_5&diff=3935&oldid=prevHelmutKnaust at 15:46, 28 October 20212021-10-28T15:46:35Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''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 $x\in D$.  </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''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 $x\in D$.  </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>#Let $f,g:<del class="diffchange diffchange-inline">(0,1)</del>\to\mathbb{R}$ be two bounded functions that are uniformly continuous on $<del class="diffchange diffchange-inline">(0,1)</del>$. Show that $f\cdot g$ is uniformly continuous on $<del class="diffchange diffchange-inline">(0,1)</del>$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>#Let $f,g:<ins class="diffchange diffchange-inline">\mathbb{R}</ins>\to\mathbb{R}$ be two bounded functions that are uniformly continuous on $<ins class="diffchange diffchange-inline">\matbb{R}</ins>$. Show that $f\cdot g$ is uniformly continuous on $<ins class="diffchange diffchange-inline">\mathbb{R}</ins>$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that the result may fail without the boundedness condition.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that the result may fail without the boundedness condition.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#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 $[0,1]$.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#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 $[0,1]$.</div></td></tr>
</table>HelmutKnausthttp://helmut.knaust.info/mediawiki/index.php?title=CRN_11378:_HW_5&diff=3934&oldid=prevHelmutKnaust at 15:43, 28 October 20212021-10-28T15:43:43Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that the result in 2. above may fail if $c=1$.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that the result in 2. above may fail if $c=1$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''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 $x\in D$.  </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Problem 24.''' Let $D\subseteq\mathbb{R}<ins class="diffchange diffchange-inline">$</ins>. A function $f:D \to \mathbb{R}$ is called ''bounded'' if there is an $M>0$ such that $|f(x)|\leq M$ for all $x\in D$.  </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>#Let $f,g:<del class="diffchange diffchange-inline">\mathbb{R}</del>\to\mathbb{R}$ be two bounded functions that are uniformly continuous on $<del class="diffchange diffchange-inline">\mathbb{R}</del>$. Show that $f\cdot g$ is uniformly continuous on $<del class="diffchange diffchange-inline">\mathbb{R}</del>$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>#Let $f,g:<ins class="diffchange diffchange-inline">(0,1)</ins>\to\mathbb{R}$ be two bounded functions that are uniformly continuous on $<ins class="diffchange diffchange-inline">(0,1)</ins>$. Show that $f\cdot g$ is uniformly continuous on $<ins class="diffchange diffchange-inline">(0,1)</ins>$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that the result may fail without the boundedness condition.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that the result may fail without the boundedness condition.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#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 $[0,1]$.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#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 $[0,1]$.</div></td></tr>
</table>HelmutKnausthttp://helmut.knaust.info/mediawiki/index.php?title=CRN_11378:_HW_5&diff=3933&oldid=prevHelmutKnaust at 15:41, 28 October 20212021-10-28T15:41:47Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that the result in 2. above may fail if $c=1$.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that the result in 2. above may fail if $c=1$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Problem 24.''' <del class="diffchange diffchange-inline">A function </del>$<del class="diffchange diffchange-inline">f:</del>\mathbb{R} \to \mathbb{R}$ is called ''bounded'' if there is an $M>0$ such that $|f(x)|\leq M$ for all $x\in<del class="diffchange diffchange-inline">\mathbb{R}</del>$.  </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Problem 24.''' <ins class="diffchange diffchange-inline">Let </ins>$<ins class="diffchange diffchange-inline">D\subseteq</ins>\mathbb{R}<ins class="diffchange diffchange-inline">. A function $f:D </ins>\to \mathbb{R}$ is called ''bounded'' if there is an $M>0$ such that $|f(x)|\leq M$ for all $x\in <ins class="diffchange diffchange-inline">D</ins>$.  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#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 $\mathbb{R}$.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#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 $\mathbb{R}$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that the result may fail without the boundedness condition.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that the result may fail without the boundedness condition.</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">#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 $[0,1]$.</ins></div></td></tr>
</table>HelmutKnausthttp://helmut.knaust.info/mediawiki/index.php?title=CRN_11378:_HW_5&diff=2812&oldid=prevHelmutKnaust at 21:55, 5 November 20192019-11-05T21:55:05Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''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 $x,y\in\mathbb{R}$.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''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 $x,y\in\mathbb{R}$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that $f$ is uniformly continuous on $\mathbb{R}$.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that $f$ is uniformly continuous on $\mathbb{R}$.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>#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 $y,f(y),f(f(y)),f(f(f(y)))\ldots$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>#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 $y,f(y),f(f(y)),f(f(f(y)))<ins class="diffchange diffchange-inline">,</ins>\ldots$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that the result in 2. above may fail if $c=1$.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that the result in 2. above may fail if $c=1$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
</table>HelmutKnausthttp://helmut.knaust.info/mediawiki/index.php?title=CRN_11378:_HW_5&diff=2804&oldid=prevHelmutKnaust at 16:29, 5 November 20192019-11-05T16:29:44Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''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 $x\in\mathbb{R}$.  </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''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 $x\in\mathbb{R}$.  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#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 $\mathbb{R}$.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#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 $\mathbb{R}$.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>#Show that the result <del class="diffchange diffchange-inline">fails </del>without the boundedness condition.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>#Show that the result <ins class="diffchange diffchange-inline">may fail </ins>without the boundedness condition.</div></td></tr>
</table>HelmutKnausthttp://helmut.knaust.info/mediawiki/index.php?title=CRN_11378:_HW_5&diff=2803&oldid=prevHelmutKnaust at 16:27, 5 November 20192019-11-05T16:27:09Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''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 $x,y\in\mathbb{R}$.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''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 $x,y\in\mathbb{R}$.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>#<del class="diffchange diffchange-inline">Now assume that $0<c<1$. </del>Show that $f$ is uniformly continuous on $\mathbb{R}$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>#Show that $f$ is uniformly continuous on $\mathbb{R}$.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>#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 $y,f(y),f(f(y)),f(f(f(y)))\ldots$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>#<ins class="diffchange diffchange-inline">Now assume that $0<c<1$. </ins>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 $y,f(y),f(f(y)),f(f(f(y)))\ldots$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that the result in 2. above may fail if $c=1$.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that the result in 2. above may fail if $c=1$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
</table>HelmutKnausthttp://helmut.knaust.info/mediawiki/index.php?title=CRN_11378:_HW_5&diff=2802&oldid=prevHelmutKnaust at 16:26, 5 November 20192019-11-05T16:26:41Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''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.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''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.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''Problem 23.''' Let $0<del class="diffchange diffchange-inline"><c<1</del>$. Assume $f:\mathbb{R}\to\mathbb{R}$ satisfies $|f(x)-f(y)|\leq c\cdot |x-y|$ for all $x,y\in\mathbb{R}$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Problem 23.''' Let $<ins class="diffchange diffchange-inline">c\geq </ins>0$. Assume $f:\mathbb{R}\to\mathbb{R}$ satisfies $|f(x)-f(y)|\leq c\cdot |x-y|$ for all $x,y\in\mathbb{R}$.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>#Show that $f$ is uniformly continuous on $\mathbb{R}$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>#<ins class="diffchange diffchange-inline">Now assume that $0<c<1$. </ins>Show that $f$ is uniformly continuous on $\mathbb{R}$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#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 $y,f(y),f(f(y)),f(f(f(y)))\ldots$.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#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 $y,f(y),f(f(y)),f(f(f(y)))\ldots$.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>#Show that the result above may fail if $c=1$.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>#Show that the result <ins class="diffchange diffchange-inline">in 2. </ins>above may fail if $c=1$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''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 $x\in\mathbb{R}$.  </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''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 $x\in\mathbb{R}$.  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#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 $\mathbb{R}$.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#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 $\mathbb{R}$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that the result fails without the boundedness condition.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that the result fails without the boundedness condition.</div></td></tr>
</table>HelmutKnausthttp://helmut.knaust.info/mediawiki/index.php?title=CRN_11378:_HW_5&diff=2801&oldid=prevHelmutKnaust at 16:24, 5 November 20192019-11-05T16:24:50Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that $f$ is uniformly continuous on $\mathbb{R}$.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that $f$ is uniformly continuous on $\mathbb{R}$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#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 $y,f(y),f(f(y)),f(f(f(y)))\ldots$.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#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 $y,f(y),f(f(y)),f(f(f(y)))\ldots$.</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">#Show that the result above may fail if $c=1$.</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''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 $x\in\mathbb{R}$.  </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''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 $x\in\mathbb{R}$.  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#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 $\mathbb{R}$.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#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 $\mathbb{R}$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that the result fails without the boundedness condition.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that the result fails without the boundedness condition.</div></td></tr>
</table>HelmutKnausthttp://helmut.knaust.info/mediawiki/index.php?title=CRN_11378:_HW_5&diff=2800&oldid=prevHelmutKnaust at 16:22, 5 November 20192019-11-05T16:22:26Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#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 $y,f(y),f(f(y)),f(f(f(y)))\ldots$.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#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 $y,f(y),f(f(y)),f(f(f(y)))\ldots$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="background: #ffa; color:black; font-size: smaller;"><div>'''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 $x\in\mathbb{R}$.  </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Problem 24.''' A function $f:\mathbb{R} \to <ins class="diffchange diffchange-inline">\</ins>mathbb{R}$ is called <ins class="diffchange diffchange-inline">''</ins>bounded<ins class="diffchange diffchange-inline">'' </ins>if there is an $M>0$ such that <ins class="diffchange diffchange-inline">$</ins>|f(x)|\leq M$ for all $x\in\mathbb{R}$.  </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#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 $\mathbb{R}$.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#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 $\mathbb{R}$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that the result fails without the boundedness condition.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>#Show that the result fails without the boundedness condition.</div></td></tr>
</table>HelmutKnaust