Sets of range uniqueness for classes of continuous functions
نویسندگان
چکیده
In [9] 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 [3] it was shown 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 in [5] and [7] 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 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.
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