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On distributions over $\mathbb R$ whose derivatives vanishes

Why can I identify a distribution $G \in \mathcal{D}'((a,b))$, $\partial G = 0$ by a constant function?

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marked as duplicate by t.b., Davide Giraudo, Dylan Moreland, Pedro Tamaroff, Jonas Teuwen May 21 '12 at 23:36

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up vote 3 down vote accepted

Hint: fix a test function $\psi_0$ of integral $1$ and for $\phi\in \mathcal D((a,b))$, put $$f(x):=\int_a^x\phi(t)dt-\int_a^b\phi(t)dt\int_a^x\psi_0(t)dt.$$ Check that $f$ is a test function, and compute $f'$. Then compute $G(f')$ to get the wanted result.

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Did you intend $\varphi$ and $\phi$ to be the same? – Jayesh Badwaik Jun 2 '15 at 6:05
@JayeshBadwaik Yes, it is a typo. – Davide Giraudo Jun 2 '15 at 15:13

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