MATHEMATICA BOHEMICA, Vol. 141, No. 1, pp. 109-114, 2016

# Baire one functions and their sets of discontinuity

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

Jonald P. Fenecios, Department of Mathematics, Ateneo de Davao University, E. Jacinto Street, 8016 Davao, Philippines, e-mail: jpfenecios@addu.edu.ph; Emmanuel A. Cabral, Department of Mathematics, Ateneo de Manila University, Loyola Heights Campus, Katipunan Avenue, 1108 Quezon, Philippines, e-mail: ecabral@ateneo.edu; Abraham P. Racca, Department of Mathematics and Physics, Adventist University of the Philippines, Puting Kahoy, Silang, 4118 Cavite, Philippines, e-mail: abraham.racca@yahoo.com

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 $\omega_f(C_n)<\epsilon$ 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 $\epsilon$-$\delta$ 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

Classification (MSC 2010): 26A21

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