Do there exist real functions $f$ and $g$, both differentiable, such that $f^{'} g$ does not have an antiderivative?
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You could take $f(x) = x$ and $g$ to be any differentiable function that doesn't have an (elementary, I suppose?) antiderivative, say $g(x) = \exp(-x^2)$. – Watercrystal Feb 08 '21 at 21:02
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Well yes of course if $f^{'} g$ is continuous then an antiderivative exists, I'm wondering about the case when it's not. And any antiderivative, not necessarily a closed form one. – Dark Feb 08 '21 at 21:09
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Two related questions: 1. Is it necessary that every function is a derivative of some function?, 2. Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true? – LinAlg Feb 08 '21 at 23:13
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See: http://matwbn.icm.edu.pl/ksiazki/fm/fm88/fm88118.pdf – Chris Feb 09 '21 at 03:05