Let $E\subset \mathbb{R}$ and $E$ be a noncompact bounded set.
Then, there exists a limit point $x_0$ of $E$ such that $x_0 \notin E$.
Thus, $f(x) = \frac{1}{x-x_0}$ is continuous on $E$.
I can't figure out how to make $d(f(x),f(y))$ arbitrarily large, for some $x,y$ such that $d(x,y) < \delta$ for a given $\delta$.