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

Abstract

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.