Definition:Uniform Continuity/Real Numbers

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 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:

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

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.