# Continuous Mapping is Sequentially Continuous/Corollary

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## Theorem

Let $X$ and $Y$ be topological spaces.

Let $f: X \to Y$ be a mapping that is continuous at $x$.

Let $f$ be continuous (everywhere) on $X$.

Then $f$ is sequentially continuous on $X$.

## Proof

This follows immediately from the definitions:

- $(1): \quad$ A mapping is sequentially continuous everywhere in $X$ if and only if if it is sequentially continuous at each point
- $(2): \quad$ A mapping is continuous everywhere on $X$ if and only if it is continuous at each point.

$\blacksquare$