Definition:Uniform Continuity

Definition

Metric Spaces

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.

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

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x, y \in A_1: d_1 \left({x, y}\right) < \delta \implies d_2 \left({f \left({x}\right), f \left({y}\right)}\right) < \epsilon$

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

Real Numbers

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 $\vert x - y \vert < \delta$ it happens that $\left\vert{f \left({x}\right) - f \left({y}\right)}\right\vert < \epsilon$.

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

$\forall \epsilon > 0: \exists \delta > 0: \left({x, y \in I, \left |{x - y}\right| < \delta \implies \left\vert{f \left({x}\right) - f \left({y}\right)}\right\vert < \epsilon}\right)$

Also see

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 $f \left({y}\right)$ is near $f \left({x}\right)$ whenever $y$ is close to $x$. But how close you need to be in order for $\left\lvert{f \left({x}\right) - f \left({y}\right)}\right\rvert$ 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 $I$.