Past exam of the mathematics course of the University of Cambridge / 2008 / ia / Paper 1 / 11D / c / ii / Solution

Suppose $f(x)=0$ and $f(x+h)/\sqrt{|h|}\to 0$. ThenThereforeThis proves sufficiency of the second alternative even if $f$ itself is not differentiable at $x$. For example, $f(t)=|t{|}^{3/4}$ is continuous and not differentiable at zero, while $g(t)=|t{|}^{3/2}$ is differentiable there with derivative zero.