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Ranks of differentiable functions

Kechris, Alexander S. and Woodin, W. Hugh (1986) Ranks of differentiable functions. Mathematika, 33 (2). pp. 252-278. ISSN 0025-5793. doi:10.1112/S0025579300011244.

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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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Additional Information:© 1987 University College London. Received on the 1st of July, 1985. Research partially supported by a U.S. NSF Grant.
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Subject Keywords:03E15; Mathematical Logic and Foundations; Set Theory; Descriptive Set Theory
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MathSciNet ReviewMR0882498
Issue or Number:2
Classification Code:MSC: 03E15
Record Number:CaltechAUTHORS:20130528-145231519
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:38698
Deposited By: Tony Diaz
Deposited On:06 Aug 2013 17:49
Last Modified:09 Nov 2021 23:39

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