[BlackFly]

> Let f ... be Riemann integrable and F ... differentiable.

What many people don't notice the first time they read this in the fundamental theorem of Calculus is that this is a double criteria. That f needs to be integrable seems like an extraneous point when F is differentiable. This holds also for the Lebesgue integral. The understanding is usually that if F is differentiable then its derivative is integrable, that is, people understand the integral as an anti-derivative but the Riemann/Lebesgue integral version of the fundamental theorem of calculus only proves that if the function you want the anti-derivative of is integrable, so you have this separate requirement to prove that f is integrable having already proven F to be differentiable (to f).

However, this theoretical (because if you aren't a mathematician you won't be bothered by this sticking point, you'll just insist that the integral is the anti-derivative when an anti-derivative exists) defect is ameliorated by the Henstock–Kurzweil integral which is (I feel) a lot easier to define and understand than the Lebesgue integral. It is practically the Riemann integral with just a minor tweak: the delta in the delta-epsilon proof is allowed to vary by location (essentially, as you approach non-integrable singularities, you tend the delta towards zero).

For the Henstock-Kurzweil integral, if F is differentiable then f is (Henstock-Kurzweil) integrable. This happens because not every derivative is Riemann or Lebesgue integrable, you need a stronger integral.

[ghighi7878]

Henstock-Kurzweil is a neat teaching trick. Often also because it shows that definition of riemann integration is not the only possible one. It leads a good motivation for lebesque later but also to of importance of spaces.

[viscousviolin]

Does it usually get taught in the undergrad maths curriculum?

[asplake]

I wasn’t taught it, but that was forty years ago