Definition 1

A real-valued function $f$, defined on a topological space $X$, is said to be lower semi-continuous (resp. upper semi-continuous) *at a point $a \in X$, if for each $h < f(a)$ [resp. each $k > f(a)$] there is a neighbourhood $V$ of $a$ such that $h < f(x)$ [resp. $k > f(x)$] for each $x \in V$.

A real-valued function is said to be lower semi-continuous (resp. upper semi-continuous) on $X$ if it is lower semi-continuous (resp. upper semi-continuous) at every point of $X$.

A real-valued function $f$ is therefore continuous at a point $a$ if and only if it is both upper and lower semi-continuous at $a$.

If $f$ is lower semi-continuous at a point, then $-f$ is upper semi-continuous at this point, and conversely; hence we may restrict ourselves in what follows, to considering properties of lower semi-continuous functions.

It is clear that a function which is lower semi-continuous on $X$ is also lower semi-continuous on every subspace of $X$.