Uniform Continuity on a Subset of the Real Numbers

Let $E\subseteq\mathbb{R}$ and let $f:E\to\mathbb{R}$. The function $f$ is said to be uniformly continuous on $E$ if for every $\varepsilon>0$ there exists $\delta>0$ such that for all $x,y\in E$, if $|x-y|<\delta$, then