# Definition:Sequential Continuity

## Definition

Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.

Let $f: S_1 \to S_2$ be a mapping.

### At a Point

Let $x \in S_1$.

Then $f$ is sequentially continuous at (the point) $x$ if and only if:

for every sequence $\sequence {x_n}_{n \mathop \in \N}$ in $T_1$ which converges to $x$, the sequence $\sequence {\map f {x_n} }_{n \mathop \in \N}$ in $T_2$ converges to $\map f x$.

### On a Domain

$f$ is sequentially continuous on $T_1$ if and only if $f$ is sequentially continuous at $x$ for every $x \in T_1$.

## Also see

• Results about sequential continuity can be found here.