Integrability. A bounded function on a compact interval [a, b] is Riemann integrable if and only if it is continuous almost everywhere (the set of its points of discontinuity has measure zero, in the sense of Lebesgue measure).

Can a function be integrable but not continuous?

Continuous functions are integrable, but continuity is not a necessary condition for integrability. As the following theorem illustrates, functions with jump discontinuities can also be integrable.

Is every integrable function is continuous?

Continuity implies integrability; if some function f(x) is continuous on some interval [a,b], then the definite integral from a to b exists. While all continuous functions are integrable, not all integrable functions are continuous.