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
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Last modified September 10, 1998.