Definition:Continuous Mapping (Metric Space)/Point/Definition 4

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Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.

Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$.

Let $a \in A_1$ be a point in $A_1$.

$f$ is continuous at (the point) $a$ (with respect to the metrics $d_1$ and $d_2$) if and only if:

for each neighborhood $N'$ of $\map f a$ in $M_2$ there exists a corresponding neighborhood $N$ of $a$ in $M_1$ such that $f \sqbrk N \subseteq N'$.

Also known as

A mapping which is continuous at $a$ with respect to $d_1$ and $d_2$ can also be referred to as $\tuple {d_1, d_2}$-continuous at $a$.

Also see