Proposition 1

A real-valued function $f$ on a topological space $X$ is lower semi-continuous if and only if, for each finite real number $k$, $\overline{f}([k, +\infty])$ [the set of all $x \in X$ such that $f(x) > k$] is an open set in $X$ [or, equivalently, $\overline{f}([-\infty, k])$ is a closed set in $X$].

For this condition shows that $\overline{f}([k, +\infty])$ is a neighbourhood of each of its points.

For $f$ to be lower semi-continuous on $X$ it is sufficient that $f^{-1}([k, +\infty])$ is open in $X$ for all real numbers $k$ belonging to a dense subset of $\mathbf{R}$.