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

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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 $A_2$.

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:

- $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in A_1: \map {d_1} {x, a} < \delta \implies \map {d_2} {\map f x, \map f a} < \epsilon$

where $\R_{>0}$ denotes the set of all strictly positive real numbers.

## Also known as

A mapping which is **continuous at $a$ with respect to $d_1$ and $d_2$** can also be referred to as **$\tuple {d_1, d_2}$-continuous at $a$**.

## Also see

## Sources

- 1953: Walter Rudin:
*Principles of Mathematical Analysis*: $4.5$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: The Definition - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 3$: Continuity: Definition $3.2$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.1$: Motivation: Definition $2.1.3$