Ranks of differentiable functions

Abstract

The purpose of this paper is to define and study a natural rank function
\nwhich associates to each differentiable function (say on the interval [0, 1]) a
\ncountable ordinal number, which measures the complexity of its derivative.
\nFunctions with continuous derivatives have the smallest possible rank 1, a
\nfunction like x^2 sin (x^(-1)) has rank 2, etc., and we show that functions of any
\ngiven countable ordinal rank exist. This exhibits an underlying hierarchical
\nstructure of the class of differentiable functions, consisting of ω_1 distinct levels.
\nThe definition of rank is invariant under addition of constants, and so it
\nnaturally assigns also to every derivative a unique rank, and an associated
\nhierarchy for the class of all derivatives.
\n
\nThe set D of functions in C[0, 1] which are everywhere differentiable is a
\ncomplete coanalytic (and thus non-Borel) set (Mazurkiewicz [Maz]; see Section
\n2 below) and it will tum out that the rank function we define has the
\nright descriptive set theoretic properties summarized in the concept of a
\ncoanalytic norm, explained in Section 1.
\n
\nOur original description of the rank function was in terms of wellfounded
\ntrees and is given in Section 4. In Section 3 we give an equivalent description
\nin terms of a Cantor-Bendixson type analysis. We would like to acknowledge
\nhere the contribution of D. Preiss. It was in a conversation with one of the
\nauthors that this equivalent description was formulated.
\n