Theorem 4

The upper envelope of a family $(f_i)$ of functions which are lower semi-continuous at a point $a \in X$ is lower semi-continuous at $a$.

Let $g$ be the upper envelope. For each $h < g(a)$ there is an index $i$ such that $h < f_i(a) \leq g(a)$, and a neighbourhood $V$ of $a$ such that $h < f_i(x)$ for all $x \in V$; hence $a$ fortiori $h < g(x)$ for all $x \in V$.

It follows from Proposition 2 that the lower envelope of a finite number of lower semi-continuous functions is again lower semi-continuous; but this is not in general true for the lower envelope of an infinite family of lower semi-continuous functions. For example, if $r$ is any rational number, let $f_r$ denote the function which is equal to 0 at $r$ and equal to 1 for all real numbers $x \neq r$; the lower envelope of the $f_r$ is the function $g$ which is equal to 0 for every rational number and 1 for every irrational number ("Dirichlet's function"), and this function is not lower semi-continuous at irrational points.