In terms of Riemann integrability: If we are taking into consideration Riemann integrals on a closed interval, then any continuous function is integrable. In terms of improper integrals: continuity does not imply integrability.

Is continuity necessary for integrability?

Continuous functions are integrable, but continuity is not a necessary condition for integrability. ... With the geometric interpretation of the integral as the area below the graph of a positive function, the last property simply states that the total area is equal to the sum of its disjoint parts.

Does Riemann integrability imply continuity?

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).