Sets of range uniqueness for classes of continuous ...



Title Sets of range uniqueness for classes of continuous functions
Author(s) Maxim R. Burke, K. Ciesielski
Journal Proceedings of the American Mathematical Society
Date 1999
Volume 127
Issue 11
Start page 3295
End page 3304
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 of R for the class C-n( R) of continuous nowhere constant functions from R to 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( 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 of R with the dual property that for any f; g is an element of C-n( R) if f(-1) [M] = g(-1) [M], then f = g.

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