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.