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I have a question about the closet theorem for multidimensional functions, that is, the most general closet theorem which holds for f:R**n -> R**m. For

I have a question about the closet theorem for multidimensional functions, that is, the most general closet theorem which holds for f:R**n -> R**m. For the closet theorem to hold for all x,y in U, U must be compact and convex, but why must U be compact? I know that it must have something to do with a proof step.

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