I a, going through chapter 2 of Rudin’s mathematical analysis. After giving the definition of compactness the authors says that notion of compactness is important in analysis especially in connection with continuity.

My intuition is that in metric space if domain is compact set than continuity implies uniform continuity. Is this correct, is there any other connection between continuity and compactness. What is the connection in non metric spaces.

@mathematicsstudent and Burgo Thanks for the correction, I typed absolute continuity by mistake actually I was thinking about uniform continuity

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    $\begingroup$ There are many counterexamples on compact subsets of the reals. Absolute continuity implies differentiability almost everywhere, but there are continuous functions that are nowhere differentiable. It is true that on a compact set, continuity implies uniform continuity, but this is weaker than absolute continuity. $\endgroup$ – Bungo Dec 12 '18 at 5:14

No, this is incorrect. The standard counterexample is the Cantor function.


As said by others, absolute continuity fails.

We do have uniform continuity automatically for continuous functions on compact domains, and all continuous functions on compacta are bounded and assume their bounds.

  • $\begingroup$ Thanks henno are compactness and continuity also related in non metric spaces. I don’t think uniform continuity is defined for non metric spaces. $\endgroup$ – Curious Dec 12 '18 at 6:17
  • $\begingroup$ @Curious it is defined in uniform spaces, and there the same result holds. The boundedness holds for all real valued functions of course. $\endgroup$ – Henno Brandsma Dec 12 '18 at 16:48
  • $\begingroup$ Thanks for your response $\endgroup$ – Curious Dec 12 '18 at 16:57

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