Sets of range uniqueness for classes of continuous functions

Maxim R. Burke, Krzysztof Ciesielski
1999 Proceedings of the American Mathematical Society  
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 ⊂ R for the class Cn(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
more » ... et 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 ⊂ R with the dual property that for any f, g ∈ Cn(R) if f −1 [M] = g −1 [M], then f = g.
doi:10.1090/s0002-9939-99-04905-9 fatcat:i2rk3vutxje35jorbme2v53fii