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Title: Baire one functions and their sets of discontinuity (English)
Author: Fenecios, Jonald P.
Author: Cabral, Emmanuel A.
Author: Racca, Abraham P.
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959 (print)
ISSN: 2464-7136 (online)
Volume: 141
Issue: 1
Year: 2016
Pages: 109-114
Summary lang: English
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Category: math
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Summary: 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 $\{C_n\}_{n=1}^{\infty }$ such that $D_f=\bigcup _{n=1}^{\infty }C_n$ and $\omega _f(C_n)<\epsilon $ for each $n$ where $$ \omega _f(C_n)=\sup \{|f(x)-f(y)|\colon x,y \in C_n\} $$ 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. (English)
Keyword: Baire class one function
Keyword: set of points of discontinuity
Keyword: oscillation of a function
MSC: 26A21
idZBL: Zbl 06562163
idMR: MR3475142
DOI: 10.21136/MB.2016.9
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Date available: 2016-03-17T19:50:50Z
Last updated: 2017-06-04
Stable URL: http://hdl.handle.net/10338.dmlcz/144856
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Reference: [7] Natanson, I. P.: Theory of Functions of a Real Variable. II.Frederick Ungar Publishing New York German (1961). MR 0067952
Reference: [8] Zhao, D.: Functions whose composition with Baire class one functions are Baire class one.Soochow J. Math. 33 (2007), 543-551. Zbl 1137.26300, MR 2404581
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