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 C_n(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 C_n(R) if f^-1[M]=g^-1[M] then f=g.

Bibliography:

• [BD] A. Berarducci and D. Dikranjan, Uniformly approachable functions and UA spaces, Rend. Ist. Matematico Univ. di Trieste 25 (1993), 23--56.
• [BC] M. Burke and K. Ciesielski, Sets on which measurable functions are determined by their range, Canad. J. Math. 49 (1997) 1089-1116.
• [CS] K. Ciesielski, S. Shelah, Model with no magic set, J. Symbolic Logic 64(4) (1999), 1467-1490.
• [DPR] H. G. Diamond, C. Pomerance, L. Rubel, Sets on which an entire function is determined by its range, Math Z. 176 (1981), 383--398.

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Journal Articles

K. Ciesielski, Set Theoretic Real Analysis, J. Appl. Anal. 3(2) (1997), 143-190. MR 99k:03038

M. Burke and K. Ciesielski, Sets on which measurable functions are determined by their range, Canad. J. Math. 49 (1997), 1089-1116. MR 99i:28004

K. Ciesielski and S. Shelah, Model with no magic set, J. Symbolic Logic 64(4) (1999), 1467-1490.

Last modified January 20, 2014.