Differentiating an integral function at a point of discontinuity of the integrand

Suppose f is a real valued function which is Riemann integrable on the interval [a, b]. Let Suppose further that x 0ε(a, b), that f is continuous on a deleted neighbourhood of x 0, but that f is discontinuous at x 0. We find that if x 0 is a removable discontinuity, then F‘(x 0) exists but F‘(x o)?f(x 0), and that if x 0 is a jump discontinuity then F‘(x 0) does not exist. Finally, if x 0 is an essential discontinuity, we give examples to show that one may have F‘(x 0) = f(x 0) or that F‘(x 0) exists but F‘(x 0) ? f(x 0) or that F‘(x 0) does not exist.