Baire one functions and their sets of discontinuity
A characterization of functions in the first Baire class in terms of their sets of discontinuity is given. More precisely, a function $f \mathbb{R}\rightarrow\mathbb{R}$ 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}...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics of the Czech Academy of Science
2016-04-01
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| Series: | Mathematica Bohemica |
| Subjects: | |
| Online Access: | http://mb.math.cas.cz/full/141/1/mb141_1_9.pdf |
| Summary: | A characterization of functions in the first Baire class in terms of their sets of discontinuity is given. More precisely, a function $f \mathbb{R}\rightarrow\mathbb{R}$ 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 $ømega_f(C_n)<\epsilon$ for each $n$ where
ømega_f(C_n)=\sup\{|f(x)-f(y)| 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. |
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| ISSN: | 0862-7959 2464-7136 |