Definition:Continuous Mapping (Metric Space)/Point/Definition 2

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 $A_1$.

$f$ is continuous at (the point) $a$ (with respect to the metrics $d_1$ and $d_2$) iff:
 * $(1): \quad$ The limit of $f \left({x}\right)$ as $x \to a$ exists
 * $(2): \quad \displaystyle \lim_{x \to a} f \left({x}\right) = f \left({a}\right)$.

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

 * Equivalence of Definitions of Metric Space Continuity at Point