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Restricting us to function of a single real variable, I was used to prove Extreme value theorem via the short way: show that continuous functions preserve compactness, and the job is done.

Now, I want to prove similar results for semicontinuous functions. Wikipedia show me how to do:

This convinces me, however is a bit long and messy.

There is any way to prove the theorem more straightforwardly? For example, proving that an upper semicontinuous function has an image that owns its supremum?


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There is a more straightforward way by using that a function is lower semicontinuous if and only if for each $c \in \mathbb{R}$, the set $\{ x \in X : f(x) \le c \}$ is closed.

For the proof of this and the proof of extreme value theorem (also known as Weierstrass' theorem) using this fact, see pp.43-44 of Infinite Dimensional Analysis: Hitchhiker's Guide, which preview is available in google books.

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