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Given a topological space ##X## and two functions ##f,g:X\rightarrow \mathbb{R}##, it is rather easy to prove that ##x\rightarrow \max\{f(x),g(x)\}## is continuous. I wonder if this also holds for infinitely many functions. Of course, the maximum doesn't need to exist, so we will at least need some compactness result to let the maximum exist.

The specific form I'm talking about is to let ##C## be compact and to give a function ##\varphi:X\times C\rightarrow \mathbb{R}## (perhaps continuous or something). Then we let

[tex]x\rightarrow \max_{c\in C} \varphi(x,c).[/tex] Is this continuous?

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# Continuity of the maximum

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