Characterization of Lower Semicontinuity

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

The following are equivalent:


 * 1) $f$ is lower semicontinuous on $S$.
 * 2) The epigraph of $f$ is a closed set in $S\times\R$.
 * 3) All (lower) level sets of $f$ are closed in $S$.