Proposition 3

A real-valued function f, defined on a topological space X, is lower semi-continuous at a point $a \in X$ if and only if $\liminf_{x \to a} f(x) = f(a)$ [or, equivalently, if and only if $\liminf_{x \to a} f(x) \geq f(a)$].

The condition is necessary. For, given any $h < f(a)$, there is a neighbourhood V of $a$ such that $h < f(x)$ for all $x \in V$; therefore

(§ 5, no. 6, formulae (12)), and so $f(a) \leq \liminf_{x \to a} f(x)$. The condition is sufficient; for if it is satisfied, then for each $h < f(a)$ there is a neighbourhood V of $a$ such that $h \leq \inf_{x \in V} f(x)$, and therefore $f$ is lower semi-continuous at $a$.