Are there any cases which the First Fundamental Theorem of Calculus would fail?
First Fundamental Theorem of Calculus
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real-analysis
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1A true theorem never fails. – 2012-04-06
1 Answers
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More simply take $f(x)=\cases{-1,& $0\le x<1/2$\cr 1,& $1/2\le x\le 1$}$. Then F'(1/2) does not exist (write down the definition of the derivative as a limit of a difference quotient to see this).
For your example, I don't quite follow your argument, but it seems you've mentioned the necessary ingredients: $F$ is identically $0$ (this essentially follows from the proof that your $f$ is integrable), so F' is identically $0$. But F'(x)\ne f(x) for any rational number $x$.