On the Ultimate Peano Derivative

A function f : RªR is said to have an nth Generalized Peano Derivative GPD.at x if f is continuous in a neighborhood of x and there exists an integer kG0 such that the kth primitive of f has a kqn.th Peano derivative at x. An example shows that sometimes no such k exists. In this case, C.-M. Lee has proposed a further generalization when the sequence of derivates, indexed by k, converges to a common value. This value is termed the nth Ultimate Peano Derivative UPD.at x. Here we show that these generalizations of the Peano derivative are related to a certain Laplace integral for which the Tauberian theorem shows that any finite UPD is in fact a GPD.