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

 

Whitney's example by way of Assouad's embedding


Author: Piotr Hajlasz
Journal: Proc. Amer. Math. Soc. 131 (2003), 3463-3467
MSC (2000): Primary 26B05; Secondary 26B35, 28A80
Published electronically: February 6, 2003
MathSciNet review: 1991757
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Abstract: In this note we show how to use the Assouad embedding theorem (about almost bi-Lipschitz embeddings) to construct examples of $C^m$ functions which are not constant on a critical set homeomorphic to the $n$-dimensional cube. This extends the famous example of Whitney. Our examples are shown to be sharp.


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Additional Information

Piotr Hajlasz
Affiliation: Institute of Mathematics, Warsaw University, ul. Banacha 2, 02–097 Warszawa, Poland
Email: hajlasz@mimuw.edu.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-03-06913-2
PII: S 0002-9939(03)06913-2
Keywords: Critical set, Whitney's example, Whitney's extension theorem, Van Koch snowflake, Assouad's embedding
Received by editor(s): October 16, 2001
Received by editor(s) in revised form: May 29, 2002
Published electronically: February 6, 2003
Additional Notes: This work was supported by the KBN grant no.\ 2 PO3A 028 22.
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2003 American Mathematical Society