The Continuous Extension Theorem - Revision history

Khanh at 17:05, 6 November 2021

2021-11-06T17:05:47Z

Lila at 18:51, 20 October 2021

2021-10-20T18:51:50Z

Lila at 18:51, 20 October 2021

2021-10-20T18:51:03Z

Lila at 18:49, 20 October 2021

2021-10-20T18:49:36Z

Lila at 18:49, 20 October 2021

2021-10-20T18:49:04Z

Lila at 18:48, 20 October 2021

2021-10-20T18:48:22Z

Lila at 18:46, 20 October 2021

2021-10-20T18:46:05Z

Lila at 18:45, 20 October 2021

2021-10-20T18:45:02Z

Lila at 18:43, 20 October 2021

2021-10-20T18:43:05Z

Lila at 18:41, 20 October 2021

2021-10-20T18:41:51Z

← Older revision		Revision as of 17:05, 6 November 2021
Line 48:		Line 48:


−	==~~Resources~~==	+	== Licensing ==
−	* [http://mathonline.wikidot.com/the-continuous-extension-theorem The Continuous Extension Theorem], mathonline.wikidot.com	+	Content obtained and/or adapted from:
		+	* [http://mathonline.wikidot.com/the-continuous-extension-theorem The Continuous Extension Theorem, mathonline.wikidot.com] under a CC BY-SA license

@@ Line 36: / Line 36: @@
 </ul>
-:: <math>\lim_{n \to \infty} f(y_n) = \lim_{n \to \infty} [f(y_n) - f(x_n) + f(x_n)]</math>
+::: <math>\lim_{n \to \infty} f(y_n) = \lim_{n \to \infty} [f(y_n) - f(x_n) + f(x_n)]</math>
-:: <math>\lim_{n \to \infty} f(y_n) = \lim_{n \to \infty} [f(y_n) - f(x_n) ] + \lim_{n \to \infty} f(x_n)</math>
+::: <math>\lim_{n \to \infty} f(y_n) = \lim_{n \to \infty} [f(y_n) - f(x_n) ] + \lim_{n \to \infty} f(x_n)</math>
-:: <math>\lim_{n \to \infty} f(y_n) = 0 + L = L</math>
+::: <math>\lim_{n \to \infty} f(y_n) = 0 + L = L</math>
 <ul>

@@ Line 36: / Line 36: @@
 </ul>
-: <math>\lim_{n \to \infty} f(y_n) = \lim_{n \to \infty} [f(y_n) - f(x_n) + f(x_n)]</math>
+:: <math>\lim_{n \to \infty} f(y_n) = \lim_{n \to \infty} [f(y_n) - f(x_n) + f(x_n)]</math>
-: <math>\lim_{n \to \infty} f(y_n) = \lim_{n \to \infty} [f(y_n) - f(x_n) ] + \lim_{n \to \infty} f(x_n)</math>
+:: <math>\lim_{n \to \infty} f(y_n) = \lim_{n \to \infty} [f(y_n) - f(x_n) ] + \lim_{n \to \infty} f(x_n)</math>
-: <math>\lim_{n \to \infty} f(y_n) = 0 + L = L</math>
+:: <math>\lim_{n \to \infty} f(y_n) = 0 + L = L</math>
 <ul>

@@ Line 35: / Line 35: @@
 <li>Now suppose that <math>(y_n)</math> is another sequence in <math>(a, b)</math> that converges to <math>a</math>. Then <math>\lim_{n \to \infty} (x_n - y_n) = a - a = 0</math>, and so by the uniform continuity of <math>f</math>:</li>
 </ul>
 : <math>\lim_{n \to \infty} f(y_n) = \lim_{n \to \infty} [f(y_n) - f(x_n) + f(x_n)]</math>

@@ Line 36: / Line 36: @@
 </ul>
 <math>
-: \lim_{n \to \infty} f(y_n) = \lim_{n \to \infty} [f(y_n) - f(x_n) + f(x_n)]
+: <math>\lim_{n \to \infty} f(y_n) = \lim_{n \to \infty} [f(y_n) - f(x_n) + f(x_n)]</math>
-: \lim_{n \to \infty} f(y_n) = \lim_{n \to \infty} [f(y_n) - f(x_n) ] + \lim_{n \to \infty} f(x_n)
+: <math>\lim_{n \to \infty} f(y_n) = \lim_{n \to \infty} [f(y_n) - f(x_n) ] + \lim_{n \to \infty} f(x_n)</math>
-: \lim_{n \to \infty} f(y_n) = 0 + L = L</math>
+: <math>\lim_{n \to \infty} f(y_n) = 0 + L = L</math>
 <ul>

@@ Line 36: / Line 36: @@
 </ul>
-<math>\begin{align} \lim_{n \to \infty} f(y_n) = \lim_{n \to \infty} [f(y_n) - f(x_n) + f(x_n)] \\ \lim_{n \to \infty} f(y_n) = \lim_{n \to \infty} [f(y_n) - f(x_n) ] + \lim_{n \to \infty} f(x_n) \\ \lim_{n \to \infty} f(y_n) = 0 + L = L \end{align}</math>
+<math>
 <ul>
 <li>So for every sequence <math>(y_n)</math> in <math>(a, b)</math> that converges to <math>a</math>, we have that <math>(f(y_n))</math> converges to <math>L</math>. Therefore by the Sequential Criterion for Limits, we have that <math>f</math> has the limit <math>L</math> at the point <math>a</math>. Therefore, define <math>f(a) = L</math> and so <math>f</math> is continuous at <math>a</math>. We use the same argument for the endpoint <math>b</math>, and so <math>f</math> is can be extended so that <math>f</math> is continuous on <math>[a, b]</math>.</li>

@@ Line 3: / Line 3: @@
 <table class="wiki-content-table">
 <tr>
 <blockquote style="background: white; border: 1px solid black; padding: 1em;">
-<td><strong>Lemma 1:</strong> If <math>f : A \to \mathbb{R}</math> is a uniformly continuous function and if <math>(x_n)</math> is a Cauchy Sequence from <math>A</math>, then <math>(f(x_n))</math> is a Cauchy sequence from <math>\mathbb{R}</math>.</td>
+<strong>Lemma 1:</strong> If <math>f : A \to \mathbb{R}</math> is a uniformly continuous function and if <math>(x_n)</math> is a Cauchy Sequence from <math>A</math>, then <math>(f(x_n))</math> is a Cauchy sequence from <math>\mathbb{R}</math>.
 </blockquote>
 </tr>
 </table>
@@ Line 20: / Line 22: @@
 <table class="wiki-content-table">
 <tr>
 <blockquote style="background: white; border: 1px solid black; padding: 1em;">
-<td><strong>Theorem 1 (The Continuous Extension Theorem):</strong> If <math>I = (a,b)</math> is an interval, then <math>f : I \to \mathbb{R}</math> is a uniformly continuous function on <math>I</math> if and only if <math>f</math> can be defined at the endpoints <math>a</math> and <math>b</math> such that <math>f</math> is continuous on <math>[a, b]</math>.</td>
+<strong>Theorem 1 (The Continuous Extension Theorem):</strong> If <math>I = (a,b)</math> is an interval, then <math>f : I \to \mathbb{R}</math> is a uniformly continuous function on <math>I</math> if and only if <math>f</math> can be defined at the endpoints <math>a</math> and <math>b</math> such that <math>f</math> is continuous on <math>[a, b]</math>.
 </blockquote>
 </tr>
 </table>

@@ Line 27: / Line 27: @@
 <li>Now suppose that <math>(y_n)</math> is another sequence in <math>(a, b)</math> that converges to <math>a</math>. Then <math>\lim_{n \to \infty} (x_n - y_n) = a - a = 0</math>, and so by the uniform continuity of <math>f</math>:</li>
 </ul>
-<span class="equation-number">(1)
+<span class="equation-number">
 <math>\begin{align} \lim_{n \to \infty} f(y_n) = \lim_{n \to \infty} [f(y_n) - f(x_n) + f(x_n)] \\ \lim_{n \to \infty} f(y_n) = \lim_{n \to \infty} [f(y_n) - f(x_n) ] + \lim_{n \to \infty} f(x_n) \\ \lim_{n \to \infty} f(y_n) = 0 + L = L \end{align}</math>
 <ul>