Ranks of differentiable functions

Abstract

The purpose of this paper is to define and study a natural rank function which associates to each differentiable function (say on the interval [0, 1]) a countable ordinal number, which measures the complexity of its derivative. Functions with continuous derivatives have the smallest possible rank 1, a function like x^2 sin (x^(-1)) has rank 2, etc., and we show that functions of any given countable ordinal rank exist. This exhibits an underlying hierarchical structure of the class of differentiable functions, consisting of ω_1 distinct levels. The definition of rank is invariant under addition of constants, and so it naturally assigns also to every derivative a unique rank, and an associated hierarchy for the class of all derivatives. The set D of functions in C[0, 1] which are everywhere differentiable is a complete coanalytic (and thus non-Borel) set (Mazurkiewicz [Maz]; see Section 2 below) and it will tum out that the rank function we define has the right descriptive set theoretic properties summarized in the concept of a coanalytic norm, explained in Section 1. Our original description of the rank function was in terms of wellfounded trees and is given in Section 4. In Section 3 we give an equivalent description in terms of a Cantor-Bendixson type analysis. We would like to acknowledge here the contribution of D. Preiss. It was in a conversation with one of the authors that this equivalent description was formulated.

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