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

Definition
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 $d_2$) :
 * $(1): \quad$ The limit of $\map f x$ as $x \to a$ exists
 * $(2): \quad \ds \lim_{x \mathop \to a} \map f x = \map f a$.

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

 * Equivalence of Definitions of Metric Space Continuity at Point