https://drorbn.net/index.php?action=history&feed=atom&title=1617-257%2FTUT-R-81617-257/TUT-R-8 - Revision history2026-06-21T10:04:21ZRevision history for this page on the wikiMediaWiki 1.39.6https://drorbn.net/index.php?title=1617-257/TUT-R-8&diff=15634&oldid=prevJeffim at 20:36, 7 November 20162016-11-07T20:36:27Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:36, 7 November 2016</td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Problem 2'''. Let <math>f: B_1(0) \to \mathbb{R}^2</math> be a function which is "jelly-rigid": for all <math>x,y \in B_1(0)</math>: <math>|f(x) - f(y) - (x - y)| \leq 0.1 |x - y|</math>. Prove that <math>f</math> maps onto <math>B_{0.4}(0)</math>.</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Problem 2'''. Let <math>f: B_1(0) \to \mathbb{R}^2</math> be a function which is "jelly-rigid": for all <math>x,y \in B_1(0)</math>: <math>|f(x) - f(y) - (x - y)| \leq 0.1 |x - y|</math>. Prove that <math>f</math> maps onto <math>B_{0.4}(0)</math>.</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="color: #202122; 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>''Proof''. Since <math>f</math> is Lipschitz, it has a unique extension to its boundary and so we regard it as a function on <math>B:= \overline{B_1(0)}</math>. A simple estimate shows if <math>f(x) \in B_{0.4}(0)</math>, then <math>x \in B_1(0)</math>. Suppose now that there is some point <math>z \in B_{0.4}(0)</math> which is not in the image of <math>f</math>. Let <math>x_0 \in <del style="font-weight: bold; text-decoration: none;">B</del></math> be a closest element to <math>z</math> in the image of <math>f</math>. Consider now the point <math>x_1 := x_0 + \delta (z - f(x_0))</math> (jelly-rigidity says that the function is almost like the identity, so moving closer to the point <math>z</math> from <math>f(x_0)</math> in the codomain side can be obtained by moving in that same direction on the domain side first) where <math>\delta > 0</math> is chosen to be small enough so that <math>x_1 \in B_1(0)</math> and <math>0< \delta < 1</math>. Then <math>f(x_1)</math> is closer to <math>z</math> than is <math>f(x_0)</math>.</div></td>
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<td style="color: #202122; 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>''Proof''. Since <math>f</math> is Lipschitz, it has a unique extension to its boundary and so we regard it as a function on <math>B:= \overline{B_1(0)}</math>. A simple estimate shows if <math>f(x) \in B_{0.4}(0)</math>, then <math>x \in B_1(0)</math>. Suppose now that there is some point <math>z \in B_{0.4}(0)</math> which is not in the image of <math>f</math>. Let <math>x_0 \in <ins style="font-weight: bold; text-decoration: none;">B_1(0)</ins></math> be a closest element to <math>z</math> in the image of <math>f</math>. Consider now the point <math>x_1 := x_0 + \delta (z - f(x_0))</math> (jelly-rigidity says that the function is almost like the identity, so moving closer to the point <math>z</math> from <math>f(x_0)</math> in the codomain side can be obtained by moving in that same direction on the domain side first) where <math>\delta > 0</math> is chosen to be small enough so that <math>x_1 \in B_1(0)</math> and <math>0< \delta < 1</math>. Then <math>f(x_1)</math> is closer to <math>z</math> than is <math>f(x_0)</math>.</div></td>
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</table>Jeffimhttps://drorbn.net/index.php?title=1617-257/TUT-R-8&diff=15633&oldid=prevJeffim at 20:32, 7 November 20162016-11-07T20:32:47Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:32, 7 November 2016</td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Problem 2'''. Let <math>f: B_1(0) \to \mathbb{R}^2</math> be a function which is "jelly-rigid": for all <math>x,y \in B_1(0)</math>: <math>|f(x) - f(y) - (x - y)| \leq 0.1 |x - y|</math>. Prove that <math>f</math> maps onto <math>B_{0.4}(0)</math>.</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Problem 2'''. Let <math>f: B_1(0) \to \mathbb{R}^2</math> be a function which is "jelly-rigid": for all <math>x,y \in B_1(0)</math>: <math>|f(x) - f(y) - (x - y)| \leq 0.1 |x - y|</math>. Prove that <math>f</math> maps onto <math>B_{0.4}(0)</math>.</div></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="color: #202122; 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>''Proof''. Since <math>f</math> is Lipschitz, it has a unique extension to its boundary and so we regard it as a function on <math>B:= \overline{B_1(0)}</math>. A simple estimate shows if <math>f(x) \in B_{0.4}(0)</math>, then <math>x \in B_1(0)</math>. Suppose now that there is some point <math>z \in B_{0.4}(0)</math> which is not in the image of <math>f</math>. Let <math>x_0 \in B</math> be a closest element to <math>z</math> in the image of <math>f</math>. Consider now the point <math>x_1 := x_0 + \delta (z - f(x_0))</math> (jelly-rigidity says that the function is almost like the identity, so moving closer to the point <math>z</math> from <math>f(x_0)</math> in the codomain side can be obtained by moving in that same direction on the domain side first) where <math>\delta > 0</math> is chosen to be small enough so that <math>x_1 \in B_1(0)</math> and <math>0< \delta < 1</math>. Then <math>f(x_1)</math> is<del style="font-weight: bold; text-decoration: none;"> strictly</del> closer to <math>z</math> than is <math>f(x_0)</math>.</div></td>
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<td style="color: #202122; 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>''Proof''. Since <math>f</math> is Lipschitz, it has a unique extension to its boundary and so we regard it as a function on <math>B:= \overline{B_1(0)}</math>. A simple estimate shows if <math>f(x) \in B_{0.4}(0)</math>, then <math>x \in B_1(0)</math>. Suppose now that there is some point <math>z \in B_{0.4}(0)</math> which is not in the image of <math>f</math>. Let <math>x_0 \in B</math> be a closest element to <math>z</math> in the image of <math>f</math>. Consider now the point <math>x_1 := x_0 + \delta (z - f(x_0))</math> (jelly-rigidity says that the function is almost like the identity, so moving closer to the point <math>z</math> from <math>f(x_0)</math> in the codomain side can be obtained by moving in that same direction on the domain side first) where <math>\delta > 0</math> is chosen to be small enough so that <math>x_1 \in B_1(0)</math> and <math>0< \delta < 1</math>. Then <math>f(x_1)</math> is closer to <math>z</math> than is <math>f(x_0)</math>.</div></td>
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</table>Jeffimhttps://drorbn.net/index.php?title=1617-257/TUT-R-8&diff=15632&oldid=prevJeffim at 20:32, 7 November 20162016-11-07T20:32:24Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:32, 7 November 2016</td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Problem 2'''. Let <math>f: B_1(0) \to \mathbb{R}^2</math> be a function which is "jelly-rigid": for all <math>x,y \in B_1(0)</math>: <math>|f(x) - f(y) - (x - y)| \leq 0.1 |x - y|</math>. Prove that <math>f</math> maps onto <math>B_{0.4}(0)</math>.</div></td>
<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Problem 2'''. Let <math>f: B_1(0) \to \mathbb{R}^2</math> be a function which is "jelly-rigid": for all <math>x,y \in B_1(0)</math>: <math>|f(x) - f(y) - (x - y)| \leq 0.1 |x - y|</math>. Prove that <math>f</math> maps onto <math>B_{0.4}(0)</math>.</div></td>
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<td class="diff-marker"></td>
<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br /></td>
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<td style="color: #202122; 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>''Proof''. Since <math>f</math> is Lipschitz, it has a unique extension to its boundary and so we regard it as a function on <math>B:= \overline{B_1(0)}</math>. A simple estimate shows if <math>f(x) \in B_{0.4}(0)</math>, then <math>x \in B_1(0)</math>. Suppose now that there is some point <math>z \in B_{0.4}(0)</math> which is not in the image of <math>f</math>. Let <math>x_0 \in B</math> be a closest element to <math>z</math> in the image of <math>f</math>. Consider now the point <math>x_1 := x_0 + \delta (z - f(x_0))</math> (jelly-rigidity says that the function is almost like the identity, so moving closer to the point <math>z</math> from <math>f(x_0)</math> in the codomain side can be obtained by moving in that same direction on the domain side first) where <math>\delta > 0</math> is chosen to be small enough so that <math>x_1 \in B_1(0)</math> and <math>0< \delta < 1</math>.</div></td>
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<td style="color: #202122; 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>''Proof''. Since <math>f</math> is Lipschitz, it has a unique extension to its boundary and so we regard it as a function on <math>B:= \overline{B_1(0)}</math>. A simple estimate shows if <math>f(x) \in B_{0.4}(0)</math>, then <math>x \in B_1(0)</math>. Suppose now that there is some point <math>z \in B_{0.4}(0)</math> which is not in the image of <math>f</math>. Let <math>x_0 \in B</math> be a closest element to <math>z</math> in the image of <math>f</math>. Consider now the point <math>x_1 := x_0 + \delta (z - f(x_0))</math> (jelly-rigidity says that the function is almost like the identity, so moving closer to the point <math>z</math> from <math>f(x_0)</math> in the codomain side can be obtained by moving in that same direction on the domain side first) where <math>\delta > 0</math> is chosen to be small enough so that <math>x_1 \in B_1(0)</math> and <math>0< \delta < 1<ins style="font-weight: bold; text-decoration: none;"></math>. Then <math>f(x_1)</math> is strictly closer to <math>z</math> than is <math>f(x_0)</ins></math>.</div></td>
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</table>Jeffimhttps://drorbn.net/index.php?title=1617-257/TUT-R-8&diff=15631&oldid=prevJeffim: Created page with "On 11/3/16, we discussed some questions from the exam: '''Problem 1'''. Let <math>E</math> be an infinite subset of a compact metric space <math>X</math>. Show that <math>E</..."2016-11-07T20:31:35Z<p>Created page with "On 11/3/16, we discussed some questions from the exam: '''Problem 1'''. Let <math>E</math> be an infinite subset of a compact metric space <math>X</math>. Show that <math>E</..."</p>
<p><b>New page</b></p><div>On 11/3/16, we discussed some questions from the exam:<br />
<br />
'''Problem 1'''. Let <math>E</math> be an infinite subset of a compact metric space <math>X</math>. Show that <math>E</math> has a limit point.<br />
<br />
''Proof''. If <math>E</math> has no limit points, then <math>E</math> is a closed subset of a compact space and is therefore compact in itself. Since each point of <math>E</math> is isolated, we may find for each point <math>e \in E</math> a neighborhood <math>U_e</math> such that <math> E \cap U_e = \{ e \}</math>. The collection <math>\{ U_e\}_{e \in E}</math> is an open cover of E which clearly has no finite subcover.<br />
<br />
'''Problem 2'''. Let <math>f: B_1(0) \to \mathbb{R}^2</math> be a function which is "jelly-rigid": for all <math>x,y \in B_1(0)</math>: <math>|f(x) - f(y) - (x - y)| \leq 0.1 |x - y|</math>. Prove that <math>f</math> maps onto <math>B_{0.4}(0)</math>.<br />
<br />
''Proof''. Since <math>f</math> is Lipschitz, it has a unique extension to its boundary and so we regard it as a function on <math>B:= \overline{B_1(0)}</math>. A simple estimate shows if <math>f(x) \in B_{0.4}(0)</math>, then <math>x \in B_1(0)</math>. Suppose now that there is some point <math>z \in B_{0.4}(0)</math> which is not in the image of <math>f</math>. Let <math>x_0 \in B</math> be a closest element to <math>z</math> in the image of <math>f</math>. Consider now the point <math>x_1 := x_0 + \delta (z - f(x_0))</math> (jelly-rigidity says that the function is almost like the identity, so moving closer to the point <math>z</math> from <math>f(x_0)</math> in the codomain side can be obtained by moving in that same direction on the domain side first) where <math>\delta > 0</math> is chosen to be small enough so that <math>x_1 \in B_1(0)</math> and <math>0< \delta < 1</math>.</div>Jeffim