# Composite of Continuous Mappings is Continuous/Corollary

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## Theorem

Let $T_1, T_2, T_3$ each be one of:

Let $f: T_1 \to T_2$ and $g: T_2 \to T_3$ be continuous mappings.

Then the composite mapping $g \circ f: T_1 \to T_3$ is continuous.

## Proof

These follow directly from:

$\blacksquare$

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.1$: Topological Spaces: Example $3.1.9$

- 1953: Walter Rudin:
*Principles of Mathematical Analysis*... (next): $4.7$