Definition:Lower Semicontinuous

Definition
Let $f: S \to \R \cup \left\{{-\infty, \infty}\right\}$ be an extended real valued function and $S$ be endowed with a topology $\tau$.

Then $f$ is lower semicontinuous at $\bar x \in S$ iff:


 * $\displaystyle \liminf_{x \to \bar x} f \left({x}\right) = f \left({\bar x}\right)$

where $\liminf_{x \to \bar x} f \left({x}\right)$ stands for the lower limit of $f$ at $\bar x$.

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

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