Previous |  Up |  Next

# Article

 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 . Category: math . 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 . Date available: 2016-03-17T19:50:50Z Last updated: 2020-07-01 Stable URL: http://hdl.handle.net/10338.dmlcz/144856 . Reference: [1] Bąkowska, A., Pawlak, R. J.: On some characterizations of Baire class one functions and Baire class one like functions.Tatra Mt. Math. Publ. 46 (2010), 91-106. Zbl 1224.26016, MR 2731426 Reference: [2] Bressoud, D. M.: A Radical Approach to Lebesgue's Theory of Integration.MAA Textbooks Cambridge University Press, Cambridge (2008). Zbl 1165.00001, MR 2380238 Reference: [3] Bruckner, A. M., Bruckner, J. B., Thomson, B. S.: Real Analysis.Prentice-Hall International, Upper Saddle River (1997). Zbl 0872.26001 Reference: [4] Gordon, R. A.: The Integrals of Lebesgue, Denjoy, Perron, and Henstock.Graduate Studies in Mathematics 4 American Mathematical Society, Providence (1994). Zbl 0807.26004, MR 1288751, 10.1090/gsm/004/09 Reference: [5] Kuratowski, K.: Topology. I.Academic Press, New York; Państwowe Wydawnictwo Naukowe, Warszawa; Mir, Moskva Russian (1966). Reference: [6] Lee, P.-Y., Tang, W.-K., Zhao, D.: An equivalent definition of functions of the first Baire class.Proc. Am. Math. Soc. 129 (2001), 2273-2275. Zbl 0970.26004, MR 1823909, 10.1090/S0002-9939-00-05826-3 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 .

## Files

Files Size Format View
MathBohem_141-2016-1_9.pdf 222.1Kb application/pdf View/Open

Partner of