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 said to be lower semicontinuous at $\bar{x}\in S$ if:

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

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