| 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:
|
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| 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:
|
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| Reference:
|
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| Reference:
|
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| 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 |
| . |
