# Replacing differentiation with anti-integration?

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.

• You can prove that theorem using differentiation only – famesyasd Apr 21 at 12:59
• @famesyasd link? – man and laptop Apr 21 at 13:01
• ehh, I found it on youtube, there are lectures and they are in Russian – famesyasd Apr 21 at 13:05
• 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). – Paul Sinclair Apr 21 at 21:13