Definition:Continuous Mapping (Metric Space)/Point/Definition 1

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Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ 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:

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in A_1: d_1 \left({x, a}\right) < \delta \implies d_2 \left({f \left({x}\right), f \left({a}\right)}\right) < \epsilon$

where $\R_{>0}$ denotes the set of all strictly positive real numbers.

Also known as

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

Also see