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

Jump to navigation
Jump to search

## 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:

- for each neighborhood $N'$ of $\map f a$ in $M_2$ there exists a corresponding neighborhood $N$ of $a$ in $M_1$ such that $f \sqbrk N \subseteq N'$.

## Also known as

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

## Also see

## Sources

- 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 4$: Open Balls and Neighborhoods: Theorem $4.6$