Definition 4.22.1 (Semicontinuous). Let $X$ be a topological space, $f: X \to (-\infty, \infty]$, and $g: X \to [-\infty, \infty)$, then $f$ is lower semicontinuous if for each $a \in \real$, $\bracs{f > \alpha}$ is open, and $g$ is upper semicontinuous if for each $a \in \real$, $\bracs{f < \alpha}$ is open.