In [DPR] it is 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. In [BD] it was shown that the continuum hypothesis (CH) implies the existence of a similar subset M of R for the class Cn(R) of continuous nowhere constant functions from R to R, while it follows from the results in [BC] and [CS] 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 D1 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 subset M of R with the dual property that for any f,g in Cn(R) if f-1[M]=g-1[M] then f=g.
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The following works have cited this article
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