Compactness and uniform contiuity

Question

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

Answer 1

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

Answer 2

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.