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?
This is true. One can prove something stronger with little extra effort:
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.
This is proved in Bruckner, Differentiation of Real Functions, Theorem 1.3, page 40.