Consider the function $\displaystyle f(x)=xsinx$

Now, $\displaystyle f'(x)=xcosx+sinx$, which is not bounded, so it is not uniformly continuous.

So pick $\displaystyle \epsilon = 1 $, let $\displaystyle \delta > 0$, for every $\displaystyle x,y \in \mathbb {R} , |x-y| < \delta $, we need $\displaystyle |xsinx-ysiny| \geq \epsilon $, but how should I go about that? Thanks.