real analysis - Give an example of a function uniformly differentiable with unbounded derivative

Give an example of a function $$f:U\longrightarrow \mathbb{R}$$ uniformly differentiable with unbounded derivative $f'$, $U\subset\mathbb{R}$ open set.

Any hints would be appreciated.

Answer

Take $f(x) = x^{3/2}$ on $U = [0,\infty).$ Then $f'(x) = (3/2)x^{1/2},$ which is uniformly continuous on $U.$ By the MVT, $f'$ is uniformly differentiable on $[0,\infty).$ Since $f'$ is unbounded, we have an example.