Uniform Continuity Definitions

Let \(E\) be a non-empty subset of \(\mathbb{R}\). Let \(f: E \rightarrow \mathbb{R}\). Then, \(f\) is said to be uniformly continuous on \(E\) if and only if for every \(\epsilon > 0\), there is a \(\delta > 0\) such that $$ |x - a| < \delta \quad \text{ and } \quad x,a \in E \quad \text{ imply } |f(x) - f(a)| < \epsilon. $$