# Category:Definitions/Continuous Mappings on Metric Spaces

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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 $x \in A_1$.

## Pages in category "Definitions/Continuous Mappings on Metric Spaces"

The following 15 pages are in this category, out of 15 total.

### C

- Definition:Continuous at Point of Metric Space
- Definition:Continuous at Point of Metric Space/Also known as
- Definition:Continuous Mapping (Metric Space)
- Definition:Continuous Mapping (Metric Space)/Also known as
- Definition:Continuous Mapping (Metric Space)/Point
- Definition:Continuous Mapping (Metric Space)/Point/Definition 1
- Definition:Continuous Mapping (Metric Space)/Point/Definition 2
- Definition:Continuous Mapping (Metric Space)/Point/Definition 3
- Definition:Continuous Mapping (Metric Space)/Point/Definition 4
- Definition:Continuous Mapping (Metric Space)/Space
- Definition:Continuous Mapping (Metric Space)/Space/Definition 1
- Definition:Continuous Mapping (Metric Space)/Space/Definition 2
- Definition:Continuous Mapping/Metric Subspace
- Definition:Continuous on Metric Space
- Definition:Continuous on Metric Subspace