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.