If f is continuous and defined on a compact set then it is uniformly continuous.

Question

If f is continuous and defined on a compact set then it is uniformly continuous. Given > 0 every x in the domain has a corresponding delta x. Consider the balls centered at x with radius delta x. These form an open cover of the space therefore there is a finite sub-cover. The delta you are looking for should be the minimum of this finite number of delta x's. The problem with this proof is that the delta you are looking for isn't necessarily this minimum (why not?). You fix the proof by looking at the balls centered at x with radius delta/2 instead of looking at the balls of radius delta.

Explanation / Answer

Every Lipschitz continuous map between two metric spaces is uniformly continuous. In particular, every function which is differentiable and has bounded derivative is uniformly continuous. More generally, every Hölder continuous function is uniformly continuous. Every member of a uniformly equicontinuous set of functions is uniformly continuous. The tangent function is continuous on the interval (-p/2, p/2) but is not uniformly continuous on that interval. The exponential function x ex is continuous everywhere on the real line but is not uniformly continuous on the line.

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