Sets of range uniqueness for classes of continuous functions
Authors:
Maxim R. Burke and Krzysztof Ciesielski
Journal:
Proc. Amer. Math. Soc. 127 (1999), 32953304
MSC (1991):
Primary 26A15, 54C30; Secondary 04A30, 26A46, 30D20.
Published electronically:
May 11, 1999
MathSciNet review:
1610928
Fulltext PDF Free Access
Abstract 
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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 wellbehaved 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 265066310
Email:
KCies@wvnvms.wvnet.edu
DOI:
http://dx.doi.org/10.1090/S0002993999049059
PII:
S 00029939(99)049059
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
