Example 1

If $f$ has a relative minimum at a point $a$, that is to say if there is a neighbourhood $V$ of $a$ such that, for each $x \in V$, we have $f(a) \leq f(x)$, then $f$ is lower semi-continuous at $a$. In particular, if $f(a) = -\infty$, $f$ is lower semi-continuous at $a$.