# Definition:Continuous Mapping (Metric Space)/Space/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 $A_2$.

$f$ 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 $M_1$.

By definition, this is equivalent to the continuity of $f$ with respect to the induced topologies on $A_1$ and $A_2$.

## 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**.

## Also see

## Sources

- 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: More About Continuity - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 6$: Open Sets and Closed Sets: Theorem $6.3$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.3$: Open sets in metric spaces: Proposition $2.3.13$

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- 1953: Walter Rudin:
*Principles of Mathematical Analysis*... (previous) ... (next): $4.8$