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Keywords:
Baire class one function; set of points of discontinuity; oscillation of a function
Baire class one function; set of points of discontinuity; oscillation of a function
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.
References:
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