I've been having a look at whether antiderivatives have discontinuities when the integrand is a function that is piecewise continuous, or it has a finite number of discontinuities while still being bounded.
If I were to consider a function $f(x)$ that was integrable over $[a,b]$, but not continuous (only piecewise continuous here) everywhere, would the antiderivative, $\int_{a}^{x}f(t)dt$ where $x \in[a,b]$ be continuous, or discontinuous?
I've found an answer (see below image for context) that states that "for any such function, an antiderivative always exists except possibly at the points of discontinuity." What confuses me here is the fact that it is uncertain.
Thus my question is as follows;
How would I be able to tell if the antiderivative exists or not at certain points (and how would I find what these points are) if I were given a function $f(x)$ that was integrable over $[a,b]$ but has a finite amount of discontinuities?
Thanks!
