Let $ f:[a,b]\to\mathbb{R} $ be an integrable function with a primitive function.
Define:
$ F\left(x\right)=\intop_{a}^{x}f\left(t\right)dt. $
Prove/disprove:
$F$ is differentiable and it follows that $ F'=f $
So, I'm pretty sure that this is false. Otherwise we probably wouldn't mention continious function in the fundamental theorem of calculus.
But I'm having trouble finding a counter example. Obviously we do not want a continuous function as a counter example, so we are looking for a non continuous function, but the discontinuities of the function should be essential discontinuity, because otherwise it wouldn't follow Darbo's theorem and $ f $ wouldn't has an antiderviative.
Also, we still want $ f $ to be bounded, because otherwise it wouldn't be integrable.
So I guess it should be some trigonometric function that "explodes" around its discontinuities points.
I'd like some help finding one. Thanks in advance.