Definition:Continuous Mapping (Metric Space)

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This page is about continuous mappings in the context of metric spaces. For other uses, see Definition:Continuous Mapping.

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 $A_1$.


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 $\left({A_1, d_1}\right)$ to $\left({A_2, d_2}\right)$ if and only if it is continuous at every point $x \in A_1$.


Metric Subspace

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 $Y \subseteq A_1$.

By definition, $\left({Y, d_Y}\right)$ is a metric subspace of $A_1$.


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

Then $f$ is $\left({d_Y, d_2}\right)$-continuous at $a$ if and only if:

$\forall \epsilon > 0: \exists \delta > 0: d_Y \left({x, a}\right) < \delta \implies d_2 \left({f \left({x}\right), f \left({a}\right)}\right) < \epsilon$


Similarly, $f$ is $\left({d_Y, d_2}\right)$-continuous if and only if:

$\forall a \in Y: f$ is $\left({d_Y, d_2}\right)$-continuous at $a$


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.