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.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: Exercise $1.5: 17$