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 PDF

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

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 $f=g$.

References

S. Banach, Sur une classe de fonctions continues, Fund. Math. 8 (1926), 166–172.
Tomek Bartoszyński and Haim Judah, Set theory, A K Peters, Ltd., Wellesley, MA, 1995. On the structure of the real line. MR 1350295, DOI 10.1201/9781439863466
Alessandro Berarducci and Dikran Dikranjan, Uniformly approachable functions and spaces, Proceedings of the Eleventh International Conference of Topology (Trieste, 1993), 1993, pp. 23–55 (1994) (English, with English and Italian summaries). MR 1346315
Andrew Bruckner, Differentiation of real functions, 2nd ed., CRM Monograph Series, vol. 5, American Mathematical Society, Providence, RI, 1994. MR 1274044, DOI 10.1090/crmm/005
M. Burke and K. Ciesielski, Sets on which measurable functions are determined by their range, Canad. J. Math. 49 (1997), no. 6, 1089–1116.
K. Ciesielski, Set Theory for the Working Mathematician, London Math. Soc. Student Texts 39, Cambridge Univ. Press 1997.
K. Ciesielski and S. Shelah, Model with no magic set, J. Symbolic Logic, to appear. (Preprint$^\star$ available.)
Paul Corazza, The generalized Borel conjecture and strongly proper orders, Trans. Amer. Math. Soc. 316 (1989), no. 1, 115–140. MR 982239, DOI 10.1090/S0002-9947-1989-0982239-X
Harold G. Diamond, Carl Pomerance, and Lee Rubel, Sets on which an entire function is determined by its range, Math. Z. 176 (1981), no. 3, 383–398. MR 610219, DOI 10.1007/BF01214615
Fons van Engelen, Kenneth Kunen, and Arnold W. Miller, Two remarks about analytic sets, Set theory and its applications (Toronto, ON, 1987) Lecture Notes in Math., vol. 1401, Springer, Berlin, 1989, pp. 68–72. MR 1031766, DOI 10.1007/BFb0097332
J. J. Corliss, Upper limits to the real roots of a real algebraic equation, Amer. Math. Monthly 46 (1939), 334–338. MR 4, DOI 10.1080/00029890.1939.11998880
Vasile Ene, Answers to three questions of J. Foran: “Continuous functions—a survey” [Real Anal. Exchange 2 (1976/77), no. 2, 85–103; MR0507374 (58 #22429)], Real Anal. Exchange 14 (1988/89), no. 1, 243–247. MR 988371
Arnold W. Miller, Mapping a set of reals onto the reals, J. Symbolic Logic 48 (1983), no. 3, 575–584. MR 716618, DOI 10.2307/2273449
J. W. Nienhuys and J. G. F. Thiemann, On the existence of entire functions mapping countable dense sets onto each other, Nederl. Akad. Wetensch. Proc. Ser. A 79=Indag. Math. 38 (1976), no. 4, 331–334. MR 0460638, DOI 10.1016/1385-7258(76)90073-1
S. Saks, Theory of the Integral, Warszawa-Lwów 1937.