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.

DOI:
https://doi.org/10.1090/S0002-9939-99-04905-9

Published electronically:
May 11, 1999

MathSciNet review:
1610928

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Abstract | References | Similar Articles | Additional Information

Abstract: Diamond, Pomerance and Rubel (1981) proved that there are subsets of the complex plane such that for any two entire functions and if , then . Baraducci and Dikranjan showed in 1993 that the continuum hypothesis (CH) implies the existence of a similar set for the class of continuous nowhere constant functions from to , 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 , including the class of differentiable functions and the class of absolutely continuous functions, a set with the above property can be constructed in ZFC. We will also prove the existence of a set with the dual property that for any if , then .

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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:
https://doi.org/10.1090/S0002-9939-99-04905-9

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