Sets of range uniqueness for classes of continuous functions

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.