# Category:Uniformly Continuous Functions

Jump to navigation
Jump to search

This category contains results about **Uniformly Continuous Functions**.

### Metric Spaces

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.

Then a mapping $f: A_1 \to A_2$ is **uniformly continuous on $A_1$** if and only if:

- $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x, y \in A_1: \map {d_1} {x, y} < \delta \implies \map {d_2} {\map f x, \map f y} < \epsilon$

where $\R_{>0}$ denotes the set of all strictly positive real numbers.

### Real Function

Let $I \subseteq \R$ be a real interval.

A real function $f: I \to \R$ is said to be **uniformly continuous** on $I$ if and only if:

- for every $\epsilon > 0$ there exists $\delta > 0$ such that the following property holds:
- for every $x, y \in I$ such that $\size {x - y} < \delta$ it happens that $\size {\map f x - \map f y} < \epsilon$.

Formally: $f: I \to \R$ is **uniformly continuous** if and only if the following property holds:

- $\forall \epsilon > 0: \exists \delta > 0: \paren {x, y \in I, \size {x - y} < \delta \implies \size {\map f x - \map f y} < \epsilon}$

## Subcategories

This category has the following 2 subcategories, out of 2 total.

## Pages in category "Uniformly Continuous Functions"

The following 4 pages are in this category, out of 4 total.