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

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

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 $2.6: 14$