The Continuous Extension Theorem

From Department of Mathematics at UTSA
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Recall that from <a href="/the-uniform-continuity-theorem">The Uniform Continuity Theorem</a> that if a function is a closed and bounded interval and is continuous on , then must also be uniformly continuous on . The succeeding theorem will help us determine when a function is uniformly continuous when is instead a bounded open interval.

Before we look at The Continuous Extension Theorem though, we will need to prove the following lemma.

Lemma 1: If is a uniformly continuous function and if is a <a class="newpage" href="/cauchy-sequences">Cauchy Sequence</a> from , then is a Cauchy sequence from .
  • Proof: Let be a uniformly continuous function and let be a Cauchy sequence from . We want to show that is also a Cauchy sequence. Recall that to show that is a Cauchy sequence we must show that then such that , if then .
  • Since is uniformly continuous on , then for any , such that for all where we have that .
  • Now for , since is a Cauchy sequence then such that we have that . So this will do for the sequence . So for all we have that and from the continuity of this implies that and so is a Cauchy sequence.

We are now ready to look at The Continuous Extension Theorem.

Theorem 1 (The Continuous Extension Theorem): If is an interval, then is a uniformly continuous function on if and only if can be defined at the endpoints and such that is continuous on .
  • Proof: Suppose that is uniformly continuous on . Let be a sequence in that converges to . Then since is a convergent sequence, it must also be a Cauchy sequence. By lemma 1, since is a Cauchy sequence then is also a Cauchy sequence, and so must converge in , that is for some .
  • Now suppose that is another sequence in that converges to . Then , and so by the uniform continuity of :

(1)

\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}
  • So for every sequence in that converges to , we have that converges to . Therefore by the Sequential Criterion for Limits, we have that has the limit at the point . Therefore, define and so is continuous at . We use the same argument for the endpoint , and so is can be extended so that is continuous on .
  • Suppose that is continuous on . By <a href="/the-uniform-continuity-theorem">The Uniform Continuity Theorem</a>, since is a closed and bounded interval then is uniformly continuous.


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