Sequential Continuity is Equivalent to Continuity in Metric Space/Corollary

Corollary to Sequential Continuity is Equivalent to Continuity in Metric Space

Let $\struct {X, d}$ and $\struct {Y, e}$ be metric spaces.

Let $f: X \to Y$ be a mapping.

$f$ is continuous on $X$ if and only if $f$ is sequentially continuous on $X$.

Proof

This follows immediately from the definitions:

By definition, a mapping is sequentially continuous everywhere in $X$ if and only if it is sequentially continuous at each point.

Also by definition, a mapping is continuous everywhere in $X$ if and only if it is continuous at each point.

The result follows from Sequential Continuity is Equivalent to Continuity in Metric Space.

$\blacksquare$