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Baire one functions and their sets of discontinuity. (English) Zbl 1389.26009
Authors’ abstract: 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 Baire class one if and only if for each $$\varepsilon >0$$ there is a sequence of closed sets $$\{C_n\}_{n=1}^\infty$$ such that $$D_f=\bigcup_{n=1}^{\infty}C_n$$ and $$\omega_f(C_n)<\varepsilon$$ for each $$n$$ where $\omega_f(C_n)=\sup\{| f(x)-f(y)|: x,y\in C_n\}$ and $$D_f$$ denotes the set of points of discontinuity of $$f$$. The proof of the main theorem is based on a recent $$\varepsilon$$-$$\delta$$ 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.

##### MSC:
 26A21 Classification of real functions; Baire classification of sets and functions
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