Definition:Lower Semicontinuous/Subset

Definition

Let $f: S \to \R \cup \left\{{-\infty, \infty}\right\}$ be an extended real valued function.

Let $S$ be endowed with a topology $\tau$.

Let $A \subseteq S$, and $A \ne \varnothing$.

The function $f$ is said to be lower semicontinuous on $A$ if and only if $f$ is lower semicontinuous at every $a \in A$.