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## Main Question or Discussion Point

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.

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.

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.