Theorem 3. Every Riemann integrable function is continuous almost every- where.

Are all integrable functions 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.

Can a function be integrable but not continuous?

A function does not even have to be continuous to be integrable. Consider the step function f(x)={0x≤01x>0. It is not continuous, but obviously integrable for every interval [a,b].