Theorem: Riemann Integrals of almost Continuous Functions

• If f is a bounded function defined on a closed, bounded interval [a, b] and f is continuous except at countably many points, then f is Riemann integrable.

The converse is also true:

• If f is a bounded function defined on a closed, bounded interval [a, b] and f is Riemann integrable, then f is continuous on [a, b] except possibly at countably many points.