Characterizing derivatives by preimages of sets


Krzysztof Ciesielski

Real Anal. Exchange 23 (1997-98), 553-565.

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}.

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Last modified September 10, 1998.