When thinking about the proof of the differentiability of Taylor series, I noticed that the theorem was proved by using properties of integrals. This got me thinking: To what extent can the role of differentiation in analysis be replaced by an anti-integral?

A good motive for this is that differentiation is a discontinuous operator. This is unlike the Riemann and Lebesgue integrals. For instance, the former is continuous with respect to the sup-norm. As such, theorems that involve the interchange of limits with derivatives are usually proved using properties of integrals.

Such an anti-integral is necessarily going to be multi-valued. An example of such an anti-integral is the Radon-Nikodym Derivative, which I've just looked up.

I don't know if this question is well-defined enough. Maybe some people are interested.

  • $\begingroup$ You can prove that theorem using differentiation only $\endgroup$ – famesyasd Apr 21 at 12:59
  • $\begingroup$ @famesyasd link? $\endgroup$ – man and laptop Apr 21 at 13:01
  • $\begingroup$ ehh, I found it on youtube, there are lectures and they are in Russian $\endgroup$ – famesyasd Apr 21 at 13:05
  • $\begingroup$ The anti-integral is differentiating. You cannot avoid that discontinuity in this way (you can avoid it by using a different topology on your function space). $\endgroup$ – Paul Sinclair Apr 21 at 21:13

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