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

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## 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 $d_2$)** if and only if:

- $(1): \quad$ 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 $a$**.