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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! :)

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Please explain what the subscript $+$ or $-$ sign means exactly (and maybe that helps finding the answer yourself). –  Marc van Leeuwen Nov 19 '12 at 13:19
No. Try it with $f(x)=x$ for $x<0$. –  Samuel Nov 19 '12 at 13:21
The subscript + means: $f_+(x) = \max\{f(x), 0\}$ and $f_{-}(x)=-\min \{f(x),0\}$. –  dann Nov 19 '12 at 13:36

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