# Definition:Uniform Continuity

## Definition

### 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:

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

## Also see

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### Relationship to Continuity

The property that $f$ is **uniformly continuous** on $I$ is stronger than that of being continuous on $I$.

Intuitively, continuity on an interval means that for each *fixed* point $x$ of the interval, the value of $\map f y$ is near $\map f x$ whenever $y$ is close to $x$.

But *how close* you need to be in order for $\size {\map f x - \map f y}$ to be less than a given number may depend on the point $x$ you pick on the interval.

**Uniform continuity on an interval** means that this can be chosen in a way which is *independent* of the particular point $x$.

See the proof of this fact for a more precise explanation.

### Relationship to Absolute Continuity

The property that $f$ is **uniformly continuous on $I$** is weaker than the property that $f$ is absolutely continuous on $I$.

That is, Absolutely Continuous Real Function is Uniformly Continuous.

### Compare

- The difference between convergence and uniform convergence.