# Definition:Continuous Mapping (Metric Space)/Point/Definition 4

## 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 $A_2$.

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:

- for each neighborhood $N'$ of $f \left({a}\right)$ in $M_2$ there exists a corresponding neighborhood $N$ of $a$ in $M_1$ such that $f \left[{N}\right] \subseteq N'$.

## 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 $a$**.

## Also see

## Sources

- 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 2.4$: Open Balls and Neighborhoods: Theorem $4.6$