Definition:Uniform Continuity/Real Function

Definition
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.

Also see

 * Continuous Real Function


 * Absolute Continuity