Baire one functions and their sets of discontinuity. (English) Zbl 1389.26009

Fenecios, Jonald P.; Cabral, Emmanuel A.; Racca, Abraham P.

Math. Bohem. 141, No. 1, 109-114 (2016).

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

Reviewer: Zbigniew Grande (Bydgoszcz)

MSC:

26A21

Classification of real functions; Baire classification of sets and functions

Keywords:

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