Boundedness of derivatives and anti-derivatives of holomorphic functions as a rare phenomenon

In this article we prove a general result which in particular suggests that, on a simply connected domain Ω in C, all the derivatives and anti-derivatives of the generic holomorphic function are unbounded. A similar result holds for the operator T˜N of partial sums of the Taylor expansion with center ζ∈Ω at z=0, seen as functions of the center ζ. We also discuss a universality result of these operators T˜N.