Baire one functions and their sets of discontinuity

Fenecios, Jonald P., Cabral, Emmanuel A., and Racca, Abraham P.. "Baire one functions and their sets of discontinuity." Mathematica Bohemica 141.1 (2016): 109-114. <http://eudml.org/doc/276786>.

@article{Fenecios2016,
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 the first Baire class if and only if for each $\epsilon >0$ there is a sequence of closed sets $\lbrace C_n\rbrace _\{n=1\}^\{\infty \}$ such that $D_f=\bigcup _\{n=1\}^\{\infty \}C_n$ and $\omega _f(C_n)<\epsilon $ for each $n$ where \[ \omega \_f(C\_n)=\sup \lbrace |f(x)-f(y)|\colon x,y \in C\_n\rbrace \] 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.},
author = {Fenecios, Jonald P., Cabral, Emmanuel A., Racca, Abraham P.},
journal = {Mathematica Bohemica},
keywords = {Baire class one function; set of points of discontinuity; oscillation of a function},
language = {eng},
number = {1},
pages = {109-114},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Baire one functions and their sets of discontinuity},
url = {http://eudml.org/doc/276786},
volume = {141},
year = {2016},
}

TY - JOUR
AU - Fenecios, Jonald P.
AU - Cabral, Emmanuel A.
AU - Racca, Abraham P.
TI - Baire one functions and their sets of discontinuity
JO - Mathematica Bohemica
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 141
IS - 1
SP - 109
EP - 114
AB - 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 the first Baire class if and only if for each $\epsilon >0$ there is a sequence of closed sets $\lbrace C_n\rbrace _{n=1}^{\infty }$ such that $D_f=\bigcup _{n=1}^{\infty }C_n$ and $\omega _f(C_n)<\epsilon $ for each $n$ where \[ \omega _f(C_n)=\sup \lbrace |f(x)-f(y)|\colon x,y \in C_n\rbrace \] 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.
LA - eng
KW - Baire class one function; set of points of discontinuity; oscillation of a function
UR - http://eudml.org/doc/276786
ER -