Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T21:01:35.491Z Has data issue: false hasContentIssue false
  • English
  • Français

Ranks of differentiable functions

Published online by Cambridge University Press:  26 February 2010

Alexander S. Kechris  and
W. Hugh Woodin
Alexander S. Kechris
Affiliation:
Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, U.S.A.
W. Hugh Woodin
Affiliation:
Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, U.S.A.
Get access

Extract

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 x2 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 ω1, 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.

Type
Research Article
Information
Mathematika , Volume 33 , Issue 2 , December 1986 , pp. 252 - 278
Copyright
Copyright © University College London 1986

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

AK Ajtai, M. and Kechris, A. S.. The set of continuous functions with every convergent Fourier series. To appear in Trans. Amer. Math. Soc.Google Scholar
B Bruckner, A. M.. Differentiation of real functions. Lecture Notes in Math., Vol. 659 (Springer, 1978).Google Scholar
K1 Kechris, A. S.. Sets of everywhere singular functions. Recursion Theory Week, Proc. Oberwolfach, 1984, Ed. by Ebbinghaus, H. D., Müller, G. H. and Sacks, G. E.. Lecture Notes in Math. Vol. 1141 (Springer, 1985), 233–244.Google Scholar
K2 Kechris, A. S.. Examples of sets and norms. Mimeographed notes, April 1984.Google Scholar
K-M Kechris, A. S. and Martin, D. A.. A note on universal sets for classes of countable Gs”s. Mathematika, 22 (1975), 43–45.CrossRefGoogle Scholar
M-K Martin, D. A. and Kechris, A. S.. Infinite games and effective descriptive set theory. In Analytic Sets by C. A. Rogers et al. (Academic Press, 1980).Google Scholar
Mau Mauldin, R. D.. The set of continuous nowhere differentiable functions. Pacific J. Math., 83 (1979), 199–205.CrossRefGoogle Scholar
Maz Mazurkiewicz, S.. Über die Menge der difierenzierbaren Funktionen. Fund. Math., 27 (1936), 244–249.CrossRefGoogle Scholar
Mo Moschovakis, Y. N.. Descriptive Set Theory (North-Holland, 1980).Google Scholar
Pi Piranian, G.. The set of nondifferentiability of a continuous function. Amer. Math Monthly, 73, Part II (1966), 25–61.CrossRefGoogle Scholar
RJ Rogers, C. A. and Jayne, J. E.. K-analytic sets. In Analytic Sets, Amer. Math Monthly, 73, Part II (1966), 25–61.Google Scholar
Z Zahorski, Z.. Sur 'ensemble des points de non-derivabilité d'une function continue. BulL Soc. Math. France, 74 (1946), 147–178.CrossRefGoogle Scholar