# Category:Continuous Mappings on Metric Spaces

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This category contains results about **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 $x \in A_1$.

## Subcategories

This category has the following 5 subcategories, out of 5 total.

### C

### E

### P

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

The following 27 pages are in this category, out of 27 total.

### C

- Canonical Injection into Metric Space Product with P-Product Metric is Continuous
- Cartesian Product under Chebyshev Distance of Continuous Mappings between Metric Spaces is Continuous
- Composite of Continuous Mappings at Point between Metric Spaces is Continuous at Point
- Composite of Continuous Mappings between Metric Spaces is Continuous
- Constant Function is Continuous/Metric Space
- Constant Function is Continuous/Metric Space/Proof 1
- Constant Function is Continuous/Metric Space/Proof 2
- Continuity of Mapping between Metric Spaces by Convergent Sequence
- Continuity of Mapping to Cartesian Product under Chebyshev Distance