\begin{definition}[Definition:Lower Semicontinuous/Subset]
Let $f: S \to \overline \R$ be an extended real valued function.
Let $S$ be endowed with a topology $\tau$. 
Let $A \subseteq S$, and $A \ne \O$.
The function $f$ is said to be '''lower semicontinuous on''' $A$ {{iff}} $f$ is lower semicontinuous at every $a \in A$.
Category:Definitions/Continuity
Category:Definitions/Real Analysis
\end{definition}