Let $f:[a,b]\to\mathbb{R}\cup\{-\infty\}$. I need to prove that these are equivalent
- $f$ is upper semi continuous (usc) on $[a,b]$.
- There exist step functions $f_k$ such that $f_k \to f$
- There exist decreasing continuous functions $f_k$ such that $f_k \to f$
I don't know if it is a good approach to start with a partition of the interval $[a,b]$ and take the supremum of the funcion in each one of the semi intervals (in a previous exercise, I proved that a usc function reaches its supremum in compact sets). In that case, how can I make sure $f_k \to f$. Is there a better approach? thanks in advance.