http://drorbn.net/index.php?title=1617-257/TUT-R-5&feed=atom&action=history1617-257/TUT-R-5 - Revision history2024-03-29T04:34:18ZRevision history for this page on the wikiMediaWiki 1.21.1http://drorbn.net/index.php?title=1617-257/TUT-R-5&diff=15488&oldid=prevJeffim at 15:34, 14 October 20162016-10-14T15:34:01Z<p></p>
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<td colspan='2' style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 15:34, 14 October 2016</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>On 10/13/16, we proved that if <math>U</math> is an open and convex subset of <math>\mathbb R^n</math> and</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>On 10/13/16, we proved that if <math>U</math> is an open and convex subset of <math>\mathbb R^n</math> and</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>if <math>f : U \to \mathbb R</math> is differentiable with <math>\|D f (x)\| \leq M</math> for all <<del class="diffchange diffchange-inline">nowiki</del>>x \in U</<del class="diffchange diffchange-inline">nowiki</del>><del class="diffchange diffchange-inline">,</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>if <math>f : U \to \mathbb R</math> is differentiable with <math>\|D f (x)\| \leq M</math> for all <<ins class="diffchange diffchange-inline">math</ins>>x \in U</<ins class="diffchange diffchange-inline">math</ins>></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>then we have that <math>|f(x) - f(y)| \leq M \|x - y\|</math> for all <math>x, y \in U</math>.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>then we have that <math>|f(x) - f(y)| \leq M \|x - y\|</math> for all <math>x, y \in U</math>.</div></td></tr>
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</table>Jeffimhttp://drorbn.net/index.php?title=1617-257/TUT-R-5&diff=15487&oldid=prevJeffim: Created page with "On 10/13/16, we proved that if <math>U</math> is an open and convex subset of <math>\mathbb R^n</math> and if <math>f : U \to \mathbb R</math> is differentiable with <math>\|D..."2016-10-14T15:32:16Z<p>Created page with "On 10/13/16, we proved that if <math>U</math> is an open and convex subset of <math>\mathbb R^n</math> and if <math>f : U \to \mathbb R</math> is differentiable with <math>\|D..."</p>
<p><b>New page</b></p><div>On 10/13/16, we proved that if <math>U</math> is an open and convex subset of <math>\mathbb R^n</math> and<br />
if <math>f : U \to \mathbb R</math> is differentiable with <math>\|D f (x)\| \leq M</math> for all <nowiki>x \in U</nowiki>,<br />
then we have that <math>|f(x) - f(y)| \leq M \|x - y\|</math> for all <math>x, y \in U</math>.<br />
<br />
We also proved the analogous statement if <math>f</math> is Lipschitz continuous instead of<br />
having uniformly bounded derivative.<br />
<br />
Lastly, we created a formulation for the problem if <math>U</math> is star-shaped rather<br />
than convex.</div>Jeffim