Diamond, Pomerance and Rubel (1981) proved that there are subsets
M
M
of the complex plane such that for any two entire functions
f
f
and
g
g
if
f
[
M
]
=
g
[
M
]
f[M]=g[M]
, then
f
=
g
f=g
. Baraducci and Dikranjan showed in 1993 that the continuum hypothesis (CH) implies the existence of a similar set
M
⊂
R
M\subset \mathrm {R}
for the class
C
n
(
R
)
C_n(\mathbb {R})
of continuous nowhere constant functions from
R
\mathrm {R}
to
R
\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
(
R
)
C(\mathrm {R})
, including the class
D
1
D^1
of differentiable functions and the class
A
C
AC
of absolutely continuous functions, a set
M
M
with the above property can be constructed in ZFC. We will also prove the existence of a set
M
⊂
R
M\subset \mathbb {R}
with the dual property that for any
f
,
g
∈
C
n
(
R
)
f,g\in C_n(\mathrm {R})
if
f
−
1
[
M
]
=
g
−
1
[
M
]
f^{-1}[M]=g^{-1}[M]
, then
f
=
g
f=g
.