# Definition:Continuous Mapping (Metric Space)/Space

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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 $A_2$.

### Definition 1

$f$ is **continuous from $\struct {A_1, d_1}$ to $\struct {A_2, d_2}$** if and only if it is continuous at every point $x \in A_1$.

### Definition 2

$f$ is **continuous from $\struct {A_1, d_1}$ to $\struct {A_2, d_2}$** if and only if:

- for every $U \subseteq A_2$ which is open in $M_2$, $f^{-1} \sqbrk U$ is open in $M_1$.

## Also known as

A mapping which is **continuous from $\struct {A_1, d_1}$ to $\struct {A_2, d_2}$** can also be referred to as **$\tuple {d_1, d_2}$-continuous**.