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I have a similar question to this one: Integrable or antiderivative.

If a function has an antiderivative, does the difference of values of the antiderivative on the endpoints of an interval always equal the function's integral (Riemann, Cauchy, Lebesgue) over that interval?

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Sparky
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    Yes, if the function is integrable. There are derivatives that are not Riemann integrable, or even Lebesgue integrable. If we use the Henstock-Kurzweil integral, then yes, this holds for all derivatives. – Andrés E. Caicedo May 21 '14 at 05:29
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    Could you give me an example of a function whose antiderivative behaves as I explained above? – Sparky May 21 '14 at 16:23
  • Not always here is an example link –  Aug 19 '14 at 06:26

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