Skip to main content Accessibility help
×
Home
Hostname: page-component-846f6c7c4f-rr2n5 Total loading time: 0.316 Render date: 2022-07-06T15:39:40.723Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

Article contents

Ranks of differentiable functions

Published online by Cambridge University Press:  26 February 2010

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.

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
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), 233244.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), 4345.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), 199205.CrossRefGoogle Scholar
Maz Mazurkiewicz, S.. Über die Menge der difierenzierbaren Funktionen. Fund. Math., 27 (1936), 244249.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), 2561.CrossRefGoogle Scholar
RJ Rogers, C. A. and Jayne, J. E.. K-analytic sets. In Analytic Sets, Amer. Math Monthly, 73, Part II (1966), 2561.Google Scholar
Z Zahorski, Z.. Sur 'ensemble des points de non-derivabilité d'une function continue. BulL Soc. Math. France, 74 (1946), 147178.CrossRefGoogle Scholar
9
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Ranks of differentiable functions
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Ranks of differentiable functions
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Ranks of differentiable functions
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *