Continuity at a Point

Let $E$ be a subset of $\mathbb{R}$, let $f:E\to\mathbb{R}$, and let $x_0\in E$. The function $f$ is continuous at $x_0$ if for every $\varepsilon>0$ there exists $\delta>0$ such that for every $x\in E$, if $|x-x_0|<\delta$, then $|f(x)-f(x_0)|<\varepsilon$.