Answer to Question #190651 in Real Analysis for Nikhil

Question

An integrable function can have finitely many points of discontinuties. 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 $a_1,a_2,.......,a_n$ are the points of discontinuities .

We know that continuous functions are integrable and the integral is additive in nature, so the following formula holds:

$\int_{a}^{b}f = \int_{a}^{a_1}f+ \int_{a}^{a_2}f+...........\int_{a_n}^{b}f$

If f is an integrable function, then all integrals exist, so the given problem statement is TRUE.