Lecture 22, Real Analysis, Week of 4/12-4/16 - Continuous Extensions & Continuity on Compact Sets

This video is a continuation of the previous one. We prove two theorems.

The first tells us that continuous functions which additionally preserve Cauchy sequences (so in particular any uniformly continuous or uniformly-continuous-on-bounded-subsets function) can be extended to a unique continuous function on the closure of its domain. As an immediate consequence we obtain a continuous exponential function b^x with base b.

The second tells us a situation in which continuity implies uniform continuity (the former is generally a weaker property). The hypothesis is that the domain is compact: intuitively for a given continuity challenge of ε there is for each x a δ(x) response; by compactness, finitely many of these δ(x) balls cover the domain, and this finiteness will allow for us to extract a single δ which satisfies the uniform continuity requirement. We fill in the details that make this rigorous.