Lower semicontinuous functions

Let $X$ be a metric space (or more generally a topological space). A function $f:X\to \mathbb {R} \cup \{+\infty \}$ is lower semicontinuous if

$\{x\in X:f(x)>a\}=f^{-1}\left((a,+\infty ]\right)$

is open in $X$ for all $a\in \mathbb {R}$.