If I have a very nice function as $f\in \mathcal{C}_0^\infty$, (infinitely many times differentiable with all derivatives bounded and continuous and with compact support).
Can I apply integration by parts in:
$\int |f'(x)|g(x) dx$?
by writing $|f'(x)| = (f'(x))_+ + (f'(x))_{-} \ $ ?
The point is.. is it true that $(f'(x))_{+} = (f_+(x))'$. If $f$ is so nice (make a plot) it seems to me that it shouldn't be a problem.
Thank you very much for any help! :)