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I have some problems about solving an exercise:

Prove that one function $f \colon [a,b] \to \mathbb{R}$ is lower semi-continuous if and only if, for all $x \in [a,b]$, we have $$f(x)=\sup \{g(x) \mid g \in C[a,b] \text{ and } g \le f \text{ over } [a,b] \}\;.$$

Assuming true that formula, I had no problems showing that $f^{-1}((t,\infty \ ])$ is open, using the property of supremum and the continuity of $g$'s.

I have difficulties proving the opposite implication. Beacuse $f \ge g$, we have that $f(x) \ge \sup{g(x)}$, but I am not able to show the other inequality.

Thank you.

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