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

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## Contents

## 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 $A_2$.

$f$ is **continuous from $\left({A_1, d_1}\right)$ to $\left({A_2, d_2}\right)$** if and only if:

- for every $U \subseteq A_2$ which is open in $M_2$, $f^{-1} \left[{U}\right]$ 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 $\left({A_1, d_1}\right)$ to $\left({A_2, d_2}\right)$** can also be referred to as **$\left({d_1, d_2}\right)$-continuous**.

## Also see

## Sources

- 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 2.6$: Open Sets and Closed Sets: Theorem $6.3$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{III}$: More About Continuity - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2.3$: Open sets in metric spaces: Proposition $2.3.13$

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