# Combination Theorem for Continuous Mappings/Metric Space/Difference Rule

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## Theorem

Let $M = \struct {A, d}$ be a metric space.

Let $\R$ denote the real numbers.

Let $f: M \to \R$ and $g: M \to \R$ be real-valued functions from $M$ to $\R$ which are continuous on $M$.

Then:

- $f - g$ is continuous on $M$.

## Proof

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

From Multiple Rule for Continuous Mappings on Metric Space:

- $-g$ is continuous on $M$.

From Sum Rule for Continuous Mappings on Metric Space:

- $f + \paren {-g}$ is continuous on $M$.

The result follows.

$\blacksquare$

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2$: Continuity generalized: metric spaces: Exercise $2.6: 14$