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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

A rough differentiable function


Authors: Bernd Kirchheim and Paul F.X. Müller
Journal: Proc. Amer. Math. Soc. 136 (2008), 3875-3881
MSC (2000): Primary 26A16, 30D55, 26A24, 30C99
Published electronically: June 26, 2008
MathSciNet review: 2425727
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Abstract | References | Similar Articles | Additional Information

Abstract: A real-valued continuously differentiable function $ f$ on the unit interval is constructed such that

$\displaystyle \sum_{k=1}^\infty \beta_f (x, 2^{-k} ) = \infty $

holds for every $ x \in [0,1].$ Here $ \beta_f (x, 2^{-k} )$ measures the distance of $ f$ to the best approximating linear function at scale $ 2^{-k}$ around $ x$.


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Additional Information

Bernd Kirchheim
Affiliation: Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, United Kingdom

Paul F.X. Müller
Affiliation: Institut für Analysis und Numerik, J. Kepler Universität Linz, A-4040 Linz, Austria
Email: pfxm@bayou.uni-linz.ac.at

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09322-2
PII: S 0002-9939(08)09322-2
Received by editor(s): May 17, 2002
Received by editor(s) in revised form: July 18, 2007
Published electronically: June 26, 2008
Communicated by: David Preiss
Article copyright: © Copyright 2008 by the authors