#### Title

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. .