Does a function that is differentiable at a point satisfy the Lipschitz condition around that point?

If I have a function $f(x)$ defined on a compact interval, which is differentiable at the point $x=x_0$, does it follow that it satisfies the Lipschitz condition around that point? As in, is it true that there exists a constant $C$ such that for every $x$, $$\left| f(x+x_0)-f(x_0) \right| \le C\left | x \right| $$ holds?