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:


 * $\map {\paren {f - g} } x = \map {\paren {f + \paren {-g} } } x$

From Multiple Rule for Continuous Real Functions:
 * $-g$ is ‎continuous on $S$.

From Sum Rule for Continuous Real Functions:
 * $f + \paren {-g}$ is ‎continuous on $S$.

The result follows.