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An upper semi-continuous function $f \colon X \to [-\infty,+\infty)$ defined on a topological space $X$ is always the pointwise infimum of a family of continuous functions $X \to \mathbb{R}$, and if $X$ is metric then the family of functions may be taken to be countable. This is in chapter 9 of Bourbaki's General Topology. Your result follows by the monotone convergence theorem.