# Definition:Continuous Mapping (Metric Space)/Space/Definition 1

## 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 it is continuous at every point $x \in A_1$.

## 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): $\S 2.3$: Continuity: Definition $3.2$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{III}$: The Definition - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2.1$: Motivation: Definition $2.1.3$