Baire one functions and their sets of discontinuity

• Main Authors: Fenecios, Jonald P, Cabral, Emmanuel A, Racca, Abraham P
• Format: text
• Published: Archīum Ateneo 2016
• Subjects: Mathematics
• Online Access: https://archium.ateneo.edu/mathematics-faculty-pubs/64; https://eudml.org/doc/276786
• Institution: Ateneo De Manila University

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.