In this note we will show that many classes **F**
of functions
f from **R** to **R** can be characterized by
preimages
of sets in a sense that there exist the families **A** and **D**
of subsets of **R** such that **F**=C(**D**,**A**), where

C(**D**,**A**)=
{f:**R**->**R**: f^{-1}(A) is in **D** for every A in **A**}.

In particular, we will show that there exists a Bernstein
subset B of **R** such that the
family Der of all derivatives
can be represented as Der=C(**D**,**A**), where
**A** consists of all the sets
of the form (-\infty,c), (c,\infty), and B+c with c from **R**,
and Der={g^{-1}(A): A in **A** & g in Der}.

**Requires rae.cls file**
amsmath.cls, and amssymb.cls

**DVI and
Postscript files** are available at the
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**Last modified September 10, 1998.**