# Compactness and uniform contiuity

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

• 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. – Bungo Dec 12 '18 at 5:14