A Generalization of the Riemann-Lebesgue Theorem for Riemann Integrability

Abstract

A classical theorem of Riemann and Lebesgue says that a bounded function defined on a compact interval is Riemann integrable if and only if it is continuous almost everywhere. In this note, we generalize their result and show that the difference between the upper and lower Riemann integrals of a not necessarily Riemann integrable function equals the upper Riemann integral of its oscillation function.