Proposition 2

Let $f$ and $g$ be two real-valued functions, lower semi-continuous at a point $a \in X$. Then the functions $\inf(f, g)$ and $\sup(f, g)$ are lower semi-continuous at $a$; so is $f + g$ whenever it is defined, and so is $fg$ if $f$ and $g$ are $\geq 0$ and the product $fg$ is defined.

We give the proof for $f + g$; the argument is analogous in the other cases. The result is clear if either $f(a)$ or $g(a)$ is equal to $-\infty$; if not, then $f(a) + g(a) > -\infty$. Every finite number $h < f(a) + g(a)$ can be written as $h = r + s$, where $r < f(a)$ and $s < g(a)$ are finite [it is enough to take $s$ so that $h - f(a) < s < g(a)$]; by hypothesis, there is a neighbourhood $V$ of $a$ such that, for each $x \in V$, we have $r < f(x)$, and a neighbourhood $W$ such that for each $x \in W$ we have $s < g(x)$; therefore $h = r + s < f(x) + g(x)$ for all points $x$ of the neighbourhood $V \cap W$.

In the same way we see that, if $f$ is lower semi-continuous at a point $a$, and if $f \geq 0$, then $1/f$ is upper semi-continuous at $a$.