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., (2016). Baire one functions and their sets of discontinuity. Archīum.ATENEO.
https://archium.ateneo.edu/mathematics-faculty-pubs/64