Definition:Continuous Mapping (Metric Space)/Point

Definition
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 $A_1$.

Then $f$ is continuous at (the point) $a$ (with respect to the metrics $d_1$ and $d_2$) iff:
 * $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in A_1: d_1 \left({x, a}\right) < \delta \implies d_2 \left({f \left({x}\right), f \left({a}\right)}\right) < \epsilon$

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

Also known as
A mapping which is continuous from $\left({A_1, d_1}\right)$ to $\left({A_2, d_2}\right)$ can also be referred to as $\left({d_1, d_2}\right)$-continuous.

Also see

 * Metric Space Continuity by Epsilon-Delta, where this definition is seen to be equivalent to:
 * $\forall \epsilon > 0: \exists \delta > 0: d_1 \left({x, a}\right) < \delta \implies d_2 \left({f \left({x}\right), f \left({a}\right)}\right) < \epsilon$


 * Metric Space Continuity by Open Ball, where this definition is seen to be equivalent to:
 * $\forall B_\epsilon \left({f \left({a}\right)}\right): \exists B_\delta \left({a}\right): f \left({B_\delta \left({a}\right)}\right) \subseteq B_\epsilon \left({f \left({a}\right)}\right)$