Definition:Lower Semicontinuous

Definition

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

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

Then $f$ is lower semicontinuous at $\bar x \in S$ if and only if:

$\ds \liminf_{x \mathop \to \bar x} \map f x = \map f {\bar x}$

where $\ds \liminf_{x \mathop \to \bar x} \map f x$ stands for the lower limit of $f$ at $\bar x$.

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

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

When the context is clear, the abbreviation LSC can be used to mean lower semicontinuous.