Definition:Sequential Continuity

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Definition

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

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


At a Point

Let $x \in X$.


Then $f$ is sequentially continuous at $x$ iff:

For every sequence $\left\langle{x_n}\right\rangle_{n \ge 1}$ in $X$ which converges to $x$, the sequence $\left\langle{f \left({x_n}\right)}\right\rangle_{n \ge 1}$ in $Y$ converges to $f \left({x}\right)$.


On a Domain

$f$ is sequentially continuous on $X$ iff $f$ is sequentially continuous at $x$ for every $x \in X$.