For non-empty topological spaces X and Y and arbitrary
families **A** subset of P(X) and **B** subset of P(Y) we put
$C$_{A,B}
={f:X-->Y| f[A] is in **B** for every A in **A**)}.
In this paper we will examine
which classes of functions F subset of Y^X
can be represented as $C$_{A,B}.
We will be mainly
interested in the case when
F=C(X,Y) is the class of all continuous functions from X into Y.
We prove that for non-discrete Tychonoff space X the class
F=C(X,**R**)
is not equal to $C$_{A,B}
for any
**A** subset of P(X) and **B** subset of P(**R**). Thus, C(X,**R**)
cannot be characterized by images of sets.
We also show that none of the
following classes of real functions can be represented as
$C$_{A,B}:
upper (lower) semicontinuous functions,
derivatives, approximately continuous functions,
Baire class 1 functions, Borel functions, and measurable functions.

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**Last modified September 16, 1998.**