If $f(x)$ is derivable,and $f'(x)$ is Riemann integrable, the continuity point of $f'(x)$ is dense, i.e. the measure of the set of discontinuity point of the second kind is zero. I'm thinking out that is it true that for all the derivable functions, $f'(x)$ is Riemann integrable? If not,can anyone give me a counterexample?
Assume $f$ is a function on a compact interval. A function on a compact interval is Riemann integrable if and only if it is bounded and continuous almost everywhere (the set of its points of discontinuity has measure zero, in the sense of Lebesgue measure).
Therefore what you are looking for is a function $f$ that is differentiable almost everywhere and its derivative is Lebesgue integrable but not Riemann integrable.
Take a Lebesgue set whose characteristic function is not riemann integrable. Take f to be the antiderivative of this function. Then the Lebesgue Differentiation Theorem does the rest.