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 $x$ if and only if:

For every sequence $\sequence {x_n}_{n \mathop \ge 1}$ in $X$ which converges to $x$, the sequence $\sequence {\map f {x_n} }_{n \mathop \ge 1}$ in $Y$ converges to $\map f x$.


On a Domain

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