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

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

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