A problem on the final exam is to show for a metric space (X,d) and a compact subset C in X prove that the function [tex]f(x) = min_{y \in C} d(x,y) [/tex] is continuous.(adsbygoogle = window.adsbygoogle || []).push({});

Now, there are two approches you can take. One is to go to the episolon delta definition of continuous, and the other is to use open sets.

Seeing as how C is compact, I think the better approach is to use open sets. That is, to show that for a point y = f(x), make a neighborhood around it, call it U. Then [tex]f^{-1}(U)[/tex] must be shown to be open somehow.

Taking the second approach, I can see that for any point p in [tex]f^{-1}(U)[/tex] we can construct a neighborhood V around it, so that [tex]p \subset V \subset f^{-1}(U)[/tex]. Um.. let me think... I know I can cover [tex]f^{-1}(U)[/tex] with finitely many open sets, due to the compactness of C, but I really am stuck. And the thing is, I have no idea where to begin to use the definition of f, [tex]f(x) = min_{y \in C} d(x,y) [/tex]. I'm pretty sure I'm appraoching this totally wrong but I can't think of anything else to do. Any help is greatly appreciated.

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# Show f is continuous

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