Sets of range uniqueness for classes of continuous functions

Burke, Maxim; Ciesielski, Krzysztof

doi:10.1090/S0002-9939-99-04905-9

Sets of range uniqueness for classes of continuous functions
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by Maxim R. Burke and Krzysztof Ciesielski

Proc. Amer. Math. Soc. 127 (1999), 3295-3304

DOI: https://doi.org/10.1090/S0002-9939-99-04905-9

Published electronically: May 11, 1999

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

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