The Integration by Parts Theorem states that if f' and g' are continuous, then(adsbygoogle = window.adsbygoogle || []).push({});

∫f'(x)g(x)dx = f(x)g(x) - ∫f(x)g'(x)dx.

My question is, are those assumptions necessary? For example, this holds even if only one of the functions has a continuous derivative (say f' is not continuous but g' is) since in this case the right side can be differentiated using the FTC (since f*g' is continuous) and will yield f'*g, thus being an antiderivative of f'*g.

If BOTH f' and g' are discontinuous, is there an example for which this theorem does NOT work?

I tried finding one but I couldn't... Or is it still true, but much harder to prove?

A similar question arises with the substitution rule. They assume continuity of g' in the expression f(g(x))*g'(x)

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# Integration by parts if f' ang g' are not continuous

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