1
$\begingroup$

Possible Duplicate:
Continuous function on a compact metric space is uniformly continuous

How does uniform continuity and continuity coincide in a Compact set ?

  • 2
    In fact, the question asked in the title is not the same as in the body. In the last one, we have to see whether a continuous function on a compact space is uniformly continuous, whereas in the title it's asked whether continuity implies uniform continuity. In the case of real-valued functions defined on metric spaces, we can show that a continuous function is uniformly continuous if the metric space is compact, and if each continuous function is uniformly continuous, the metric space is compact without isolated points.2012-04-26

1 Answers 1

2

Continuity is a local property. On a compact set, you can get global properties by combining a finite number of local properties.

More precisely, continuity means that given $\varepsilon>0$, for each point $x$ there is a $\delta_x>0$ such that points $\delta_x$-near $x$ are sent to points $\varepsilon$-near $f(x)$. In a compact set, you can take a finite number of $\delta_x$ to cover the domain and take $\delta>0$ as the minimum of those and so get a $\delta>0$ that works for all points for the given $\varepsilon$.

PS: I'm using the characterization of compact sets as the ones for which every open cover has a finite subcover.

  • 0
    @lhf is your answer now correct?2018-03-08