# Continuous function with Continous right derivative must be differentiable on $[0,T]$?

I think a continuous function with continuous right derivative must be differentiable on a closed bounded interval $[0,1]$ but I do not know how to prove it. If this is not true, can any one give me a counterexample?

If any one of the four Dini derivatives of a continuous function $f$ is continuous at a point then $f$ is differentiable at that point. Your hypothesis is that the upper and lower right derivatives are equal and continuous.
• Up to the right endpoint where differentiability could fail (e.g. shifting $g(x) = x \sin x$ to $f(x) = g(1-x)$). – commenter Nov 10 '12 at 21:29