# Combination Theorem for Continuous Functions/Real/Difference Rule

## Theorem

Let $\R$ denote the real numbers.

Let $f$ and $g$ be real functions which are continuous on an open subset $S \subseteq \R$.

Then:

$f - g$ is ‎continuous on $S$.

## Proof

We have that:

$\map {\paren {f - g} } x = \map {\paren {f + \paren {-g} } } x$
$-g$ is ‎continuous on $S$.
$f + \paren {-g}$ is ‎continuous on $S$.

The result follows.

$\blacksquare$