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 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 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.**