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

## 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

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

The result follows.

$\blacksquare$