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:R→R is of the first Baire class if and only if for each ϵ>0 there is a sequence of closed sets {Cn}∞n=1 such that Df=⋃∞n=1Cn and ωf(Cn)<ϵ for each n where ωf...
Saved in:
| Main Authors: | , , |
|---|---|
| 格式: | text |
| 出版: |
Archīum Ateneo
2016
|
| 主題: | |
| 在線閱讀: | https://archium.ateneo.edu/mathematics-faculty-pubs/64 https://eudml.org/doc/276786 |
| 標簽: |
添加標簽
沒有標簽, 成為第一個標記此記錄!
|
| 總結: | A characterization of functions in the first Baire class in terms of their sets of discontinuity is given. More precisely, a function f:R→R is of the first Baire class if and only if for each ϵ>0 there is a sequence of closed sets {Cn}∞n=1 such that Df=⋃∞n=1Cn and ωf(Cn)<ϵ for each n where ωf(Cn)=sup{|f(x)−f(y)|:x,y∈Cn}
and Df denotes the set of points of discontinuity of f. The proof of the main theorem is based on a recent ϵ-δ 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. |
|---|