Document Type
Article
Publication Date
1997
College/Unit
Eberly College of Arts and Sciences
Department/Program/Center
Mathematics
Abstract
In this note we will show that many classes F of real functions f : R → R can be characterized by preimages of sets in a sense that there exist families A and D of subsets of R such that F = C(D, A), where C(D, A) = {f ∈ R R : f −1 (A) ∈ D for every A ∈ A}. In particular, we will show that there exists a Bernstein B ⊂ R such that the family ∆ of all derivatives can be represented as ∆ = C(D, A), where A = S c∈R {(−∞, c),(c, ∞), B + c} and D = {g −1 (A): A ∈ A & g ∈ ∆}.
Digital Commons Citation
Ciesielski, Krzysztof, "Characterizing Derivatives by Preimages of Sets" (1997). Faculty & Staff Scholarship. 825.
https://researchrepository.wvu.edu/faculty_publications/825