Category:Definitions/Continuous Mappings on Metric Spaces

This category contains definitions related to continuous mappings in the context of metric spaces.
Related results can be found in Category:Continuous Mappings on Metric Spaces.

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 $x \in A_1$.