# Composite of Continuous Mappings between Metric Spaces is Continuous

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## Theorem

Let $M_1 = \left({X_1, d_1}\right), M_2 = \left({X_2, d_2}\right), M_3 = \left({X_3, d_3}\right)$ 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

- 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 2.3$: Continuity: Corollary $3.7$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{III}$: Path-Connectedness