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

 

Sets of range uniqueness for classes
of continuous functions


Authors: Maxim R. Burke and Krzysztof Ciesielski
Journal: Proc. Amer. Math. Soc. 127 (1999), 3295-3304
MSC (1991): Primary 26A15, 54C30; Secondary 04A30, 26A46, 30D20.
Published electronically: May 11, 1999
MathSciNet review: 1610928
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Abstract: Diamond, Pomerance and Rubel (1981) proved that there are subsets $M$ of the complex plane such that for any two entire functions $f$ and $g$ if $f[M]=g[M]$, then $f=g$. Baraducci and Dikranjan showed in 1993 that the continuum hypothesis (CH) implies the existence of a similar set $M\subset {\mathbb R}$ for the class $C_n({\mathbb R})$ of continuous nowhere constant functions from ${\mathbb R}$ to ${\mathbb R}$, while it follows from the results of Burke and Ciesielski (1997) and Ciesielski and Shelah that the existence of such a set is not provable in ZFC. In this paper we will show that for several well-behaved subclasses of $C({\mathbb R})$, including the class $D^1$ of differentiable functions and the class $AC$ of absolutely continuous functions, a set $M$ with the above property can be constructed in ZFC. We will also prove the existence of a set $M\subset {\mathbb R}$ with the dual property that for any $f,g\in C_n({\mathbb R})$ if $f^{-1}[M]=g^{-1}[M]$, then $f=g$.


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Additional Information

Maxim R. Burke
Affiliation: Department of Mathematics and Computer Science, University of Prince Edward Island, Charlottetown, Prince Edward Island, Canada C1A 4P3
Email: burke@upei.ca

Krzysztof Ciesielski
Affiliation: Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506-6310
Email: KCies@wvnvms.wvnet.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04905-9
PII: S 0002-9939(99)04905-9
Keywords: Set of range uniqueness
Received by editor(s): November 12, 1997
Received by editor(s) in revised form: February 6, 1998
Published electronically: May 11, 1999
Additional Notes: The first author’s research was supported by NSERC. The author thanks the Department of Mathematics at the University of Wisconsin for its hospitality during the year 1996/97 while much of this research was carried out.
The second author was partially supported by NATO Collaborative Research Grant CRG 950347 and 1996/97 West Virginia University Senate Research Grant.
The authors would like to thank Lee Larson for his contributions to an earlier version of the paper.
Communicated by: Alan Dow
Article copyright: © Copyright 1999 American Mathematical Society