Definition:Uniform Continuity

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

Then a mapping $f: M_1 \to M_2$ is uniformly continuous on $M_1$ if:


 * $\forall \epsilon > 0: \exists \delta > 0: \forall x, y \in M_1: d_1 \left({x, y}\right) < \delta: d_2 \left({f \left({x}\right), f \left({y}\right)}\right) < \epsilon$

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:


 * 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 $\vert f(x) - f(y) \vert < \epsilon$.

Formally: $f: I \to \R$ is uniformly continuous if the following property holds:
 * $\forall \epsilon > 0: \exists \delta > 0: \left({x, y \in I, \left |{x - y}\right| < \delta \implies \left|{f \left({x}\right) - f \left({y}\right)} \right| < \epsilon}\right)$

It can be seen that this says exactly the same thing as the definition for metric spaces if $\R$ is considered a metric space under the euclidean metric.

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 $f(y)$ is near $f(x)$ whenever $y$ is close to $x$. But how close you need to be in order for $\vert f(x) - f(y) \vert$ 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, absolute continuity implies uniform continuity.

Compare

 * The difference between convergence and uniform convergence.