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:; Emmanuel A. Cabral, Department of Mathematics, Ateneo de Manila University, Loyola Heights Campus, Katipunan Avenue, 1108 Quezon, Philippines, e-mail:; Abraham P. Racca, Department of Mathematics and Physics, Adventist University of the Philippines, Puting Kahoy, Silang, 4118 Cavite, Philippines, e-mail:

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

DOI: 10.21136/MB.2016.9

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