Composite of Continuous Mappings between Metric Spaces is Continuous

Theorem

Let $M_1 = \struct {X_1, d_1}$, $M_2 = \struct {X_2, d_2}$ and $M_3 = \struct {X_3, d_3}$ be metric spaces.

Let $f: M_1 \to M_2$ and $g: M_2 \to M_3$ be continuous mappings.

Then their composite $g \circ f: M_1 \to M_3$ is continuous.

Proof 1

From Metric Induces Topology, the metric spaces described are topological spaces.

The result follows from Composite of Continuous Mappings is Continuous.

$\blacksquare$

Proof 2

Let $f$ and $g$ be continuous mappings.

By definition:

$f$ is continuous at $a \in X_1$ for all $a \in X_1$

$g$ is continuous at $\map f a \in X_2$ for all $\map f a \in X_2$.

The result follows from Composite of Continuous Mappings at Point between Metric Spaces is Continuous at Point

$\blacksquare$

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Path-Connectedness
• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 3$: Continuity: Corollary $3.7$