\begin{definition}[Definition:Lower Semicontinuous]
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 $\bar x \in S$ {{iff}}:
:$\ds \liminf_{x \mathop \to \bar x} \map f x = \map f {\bar x}$
where $\ds \liminf_{x \mathop \to \bar x} \map f x$ stands for the lower limit of $f$ at $\bar x$.
\end{definition}