# Combination Theorem for Continuous Functions/Real/Difference Rule

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## 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 $1.5: 17$