is there exist a function, which is differentiable in [a,b] but the first derivative is not continuous for some c in (a,b)?
If a function is differentiable, then it is continuous, but in order for the derivative to exist, it must yield a finite value. If there is some point in (a,b) that causes the first derivative to be discontinuous (e.g not exist), then the function f is not differentiable at that point in [a,b]. So the answer would be no.