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Regarding the First Fundamental Thereom of Calculus why does the theorem (or at least the way it is given in the book Advanced Calculus by Fitzpatrick) only take into account continuous derivatives? Derivatives can never have jump/removable discontinuities by Darboux’s Theorem, but they can have essential discontinuities, which would mean an antiderivative would exist. Is it maybe because derivatives with essential discontinuities may have boundedness issues and thus, not be Riemann integrable - so no point in taking them into account?

funmath
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  • State the theorem you are talking about – user439545 Apr 07 '18 at 05:28
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    Most textbooks deal with the special case of continuous integrand while discussing fundamental theorem of calculus because it is simpler to deal with. But a more general version exists. Do have a look at https://math.stackexchange.com/a/1900844/72031 – Paramanand Singh Apr 07 '18 at 05:58
  • Also essential discontinuity does not necessarily mean unboundedness. Check for example the function $f(x) =\cos (1/x),x\neq 0,f(0)=0$. $f$ is bounded with only one essential discontinuity at $0$ and it's anti-derivative is $g(x) =\int_{0}^{x}f(t),dt$. – Paramanand Singh Apr 07 '18 at 06:00
  • @ParamanandSingh Thank you so much. The post you linked helped clarify everything. I still wonder why most textbooks do that. As you say it might be easier to deal with but it raises a lot of questions - and in my case, it got me confused. – funmath Apr 08 '18 at 02:44

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