Baire one functions and their sets of discontinuity

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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.