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Accumulation points of graphs of Baire-1 and Baire-2 functions. (English) Zbl 1396.26005

Baire one functions, i.e., pointwise limits of sequences of continuous functions, reveal various nice properties, which often can serve as alternative descriptions of this class of functions. The work under review contributes to this topic – the author provides a characterization of sets \(T\subset I\times\mathbb R\), \(I\subset\mathbb R\) a compact interval, that can be the set of accumulation points, \(L_f\), of some Baire one function \(f: I\to\mathbb R\) (understood as its graph). The characterization splits into two cases. First, as for bounded \(f\), it boils down to a couple of, obviously necessary, conditions on \(T\): (1) \(T\) is compact, (2) \(T(x)\), the vertical \(x\)-section of \(T\), is nonempty for every \(x\in I\), (3) the set of all \(x\in I\) with \(\#\,T(x)>1\) is a meager subset of \(I\) (Theorem 5.1). The unbounded (general) case is technically more involved since here the fact that \(\#\,L_f(x)=1\) no longer implies that \(f\) has (at worst) removable discontinuity at \(x\). The characterization conditions are as follows (Theorem 5.2): (1’) \(T\) is closed, (2’) \(T(x)\) is nonempty at nearly every \(x\in I\), (3’) the set of all \(x\in I\) with \(\#\,\overline T(x)>1\) is a meager subset of \(I\) (here, the closure \(\overline T\) of \(T\) is understood in the compactified strip \(I\times\overline{\mathbb R}\), so it may contain \(\pm\infty\)).
The same problem is considered for the class of Baire two functions (pointwise limits of sequences of Baire one functions). The characterization results (Theorems 4.1 and 4.2) say that a closed \(T\subset I\times\mathbb R\) can be of the form \(T=L_f\) for some Baire two \(f: I\to\mathbb R\) iff (1) and (2) hold (bounded case), and iff (2’) holds (general case).

MSC:

26A21 Classification of real functions; Baire classification of sets and functions
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
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