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![[parent]](http://images.planetmath.org:8080/images/uparrow.png) Viewing Message
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| ``Re: is the converse true?''
by stevecheng on 2005-07-30 14:04:25 |
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| The original question asked is quite specialized.
We might think to try to construct the F
such that F' = f by
F(x) = \int_a^x f(t) dt + F(a)
This works as long as $f$ is integrable; apparently
there might be some $f$ that are not, for example:
f(x) = d/dx [ x^2 \sin(x^{-2}) ] =
This is a standard textbook example; f is not in Lebesgue
integrable on [0,1] because of the (non-jump) discontinuity
at 0. Of course one can construct F by taking limits
and integrating from (-epsilon, 1) or something like that.
But other than this simple counterexample,
I don't know the complete answer though.
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