Fundamental Theorem of Calculus/Talk

I don't think the proof is correct: you use the mean value theorem, but that theorem only applies to differentiable functions. At that point you don't know yet that F is differentiable.

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Modern proofs do not rely on geometry. This is a faulty approach. RoseParks

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Where does geometry come into this?
Besides, proofs are either correct or incorrect. Whilst we should strive for elegance, an inelegant proof is still a proof. GWO