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 $M\subset {\mathbb R}$ for the class $C_n({\mathbb R})$ of continuous nowhere constant functions from ${\mathbb R}$ to ${\mathbb R}$, 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 $C({\mathbb R})$, 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 $M\subset {\mathbb R}$ with the dual property that for any $f,g\in C_n({\mathbb R})$ if $f^{-1}[M]=g^{-1}[M]$, then $f=g$.