Baire one functions and their sets of discontinuity
Document Type
Article
Publication Date
2016
Abstract
A characterization of functions in the first Baire class in terms of their sets of discontinuity is given. More precisely, a function f:R→R is of the first Baire class if and only if for each ϵ>0 there is a sequence of closed sets {Cn}∞n=1 such that Df=⋃∞n=1Cn and ωf(Cn)<ϵ for each n where ωf(Cn)=sup{|f(x)−f(y)|:x,y∈Cn}
and Df denotes the set of points of discontinuity of f. The proof of the main theorem is based on a recent ϵ-δ characterization of Baire class one functions as well as on a well-known theorem due to Lebesgue. Some direct applications of the theorem are discussed in the paper.
Recommended Citation
Fenecios, Jonald P., Cabral, Emmanuel A., and Racca, Abraham P.. "Baire one functions and their sets of discontinuity." Mathematica Bohemica 141.1 (2016): 109-114. .
