# Baire one functions and their sets of discontinuity

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

Mathematica Bohemica (2016)

- Volume: 141, Issue: 1, page 109-114
- ISSN: 0862-7959

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topFenecios, 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 -

## References

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