Baire one functions and their sets of discontinuity

Abstract

A characterization of functions in the first Baire class in terms of their sets of discontinuity is given. More precisely, a function $f \mathbb{R}\rightarrow\mathbb{R}$ is of the first Baire class if and only if for each $\epsilon>0$ there is a sequence of closed sets $\{C_n\}_{n=1}^{\infty}$ such that $D_f=\bigcup_{n=1}^{\infty}C_n$ and $ømega_f(C_n)<\epsilon$ for each $n$ where ømega_f(C_n)=\sup\{|f(x)-f(y)| x,y \in C_n\} and $D_f$ denotes the set of points of discontinuity of $f$. The proof of the main theorem is based on a recent $\epsilon$-$\delta$ 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.

Keywords

• Baire class one function
• set of points of discontinuity
• oscillation of a function