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Let $X$ be a compact metric space. Assume that $f: X \to \mathbb{R}$ is upper-semi continuous and $g:X \to \mathbb{R}$ is lower semi-continuous. Assume that $\sup \{ f(x)+g(x) : x \in X \}$ is finite. Is it true that $$\sup \{ f(x)+g(x) : x \in X \}$$ is attained? I doubt that the answer is positive. If so, under what minimal conditions is it true? For sure, I don't want to assume continuity. For instance, can we assume some extra conditions on the space $X$?

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    $\begingroup$ It is certainly not true. Think e.g. of $X=[0,1]$, $f=0$ identically, $g(x)=x$ for $0\le x<1$ and $g(1)=0$. $\endgroup$
    – Pietro Majer
    Commented Apr 3 at 13:33

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