Proposition 4

Let f be any real-valued function defined on a dense subset A of a topological space X. If, for each $x \in X$, we put $g(x) = \liminf_{y \to x, y \in A} f(y)$, then g is lower semi-continuous on X.

For, given any $h < g(x)$, there is an open neighbourhood V of $x$ such that, for all $z \in V \cap A$, we have $h < f(z)$; now V is a neighbourhood of each of its points $y$; thus we have $\liminf_{z \to y, z \in A} f(z) = g(y) \geq h$ for all $y \in V$, and the result follows.

The function $g$ is called the lower semi-continuous regularization of $f$. We define the upper semi-continuous regularization of $f$ similarly.

We may also define $g$ as the greatest of the lower semi-continuous functions $\varphi$ on X which are such that $\varphi(x) \leq f(x)$ for all $x \in A$. If $f$ is lower semi-continuous on A, then $g$ is an extension of $f$ to X, by Proposition 3.