# Definition:Continuous Mapping (Metric Space)

*This page is about Continuous Mapping in the context of Metric Space. For other uses, see Continuous Mapping.*

## 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 $A_1$.

### Continuous at a Point

**$f$ 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.

### Continuous on a Space

$f$ is **continuous from $\struct {A_1, d_1}$ to $\struct {A_2, d_2}$** if and only if it is continuous at every point $x \in A_1$.

## Metric Subspace

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 $Y \subseteq A_1$.

By definition, $\struct {Y, d_Y}$ is a metric subspace of $A_1$.

Let $a \in Y$ be a point in $Y$.

Then $f$ is **$\tuple {d_Y, d_2}$-continuous at $a$** if and only if:

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

Similarly, $f$ is **$\tuple {d_Y, d_2}$-continuous** if and only if:

- $\forall a \in Y: f$ is $\tuple {d_Y, d_2}$-continuous at $a$

## Also known as

A mapping which is **continuous from $\struct {A_1, d_1}$ to $\struct {A_2, d_2}$** can also be referred to as **$\tuple {d_1, d_2}$-continuous**.

## Examples

### Composition of Arbitrary Mappings

Let the following mappings be defined:

\(\ds g: \R^2 \to \R^2 \times \R^2: \, \) | \(\ds \map g {x, y}\) | \(=\) | \(\ds \tuple {\tuple {x, y}, \tuple {x, y} }\) | |||||||||||

\(\ds h: \R^2 \times \R^2 \to \R \times \R: \, \) | \(\ds \map h {\tuple {a, b}, \tuple {c, d} }\) | \(=\) | \(\ds \tuple {a + b, c - d}\) | |||||||||||

\(\ds k: \R \times \R \to \R \times \R: \, \) | \(\ds \map k {u, v}\) | \(=\) | \(\ds \tuple {u^2, v^2}\) | |||||||||||

\(\ds m: \R \times \R \to \R: \, \) | \(\ds \map k {x, y}\) | \(=\) | \(\ds \dfrac {x - y} 4\) |

where $\R$ and $\R^2$ denote the real number line and real number plane respectively, under the usual (Euclidean) metric.

Then:

- each of $g, h, k, m$ are continuous
- $x y = \map {\paren {m \circ k \circ h \circ g} } {x, y}$

where $\circ$ denotes composition of mappings.

### Identity Function with Discontinuity

Let $f: \R \to \R$ be the real function defined as:

- $\forall x \in \R: \map f x = \begin {cases} x & : x \ne 0 \\ 1 & : x = 0 \end {cases}$

Then $f$ is not continuous.