Question

An integrable function can have finitely many points of discontinuous.

True or false with full explanation

Answer 1

Let $[a,a_1], [a_2,a_3], ..., [a_n,b]$ are the intervals of continuity of the function and

$a_1, a_2, ..., a_n$ points of discontinuities. We know that continuous functions are integrable and the integral is additive, so the following formula holds:

$\int_a^bf=\int_a^{a_1}f+\int_{a_1}^{a_2}f+...+\int_{a_n}^bf$

If f is an integrable function, then all integrals exist, so the statement is true