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On the averages of Darboux functions. (English) Zbl 0908.26005
Two problems are considered by the author. (1) Characterize families \(\mathcal F\) of real functions for which there exists a Darboux function \(\psi\) such that \(\psi >f\) for each \(f\in{\mathcal F}\). (2) Characterize families \(\mathcal F\) of real functions for which there exists two Darboux functions \(\varphi,\psi\) such that \(\varphi<f<\psi\) for each \(f\in{\mathcal F}\). Note that those problems are equivalent to the following questions. \((1')\) Does there exist \(g>0\) such that \(f+g\) is Darboux for each \(f\in{\mathcal F}\)? \((2')\) Does there exist \(g>0\) such that both \(f+g\) and \(-f+g\) are Darboux for each \(f\in{\mathcal F}\)? The author answers both problems \((1')\) and \((2')\) for families \(\mathcal F\) with cardinality less than the cofinality of the continuum. Moreover, he proves that if the size of \(\mathcal F\) is less than the additivity of the measure and all \(f\in{\mathcal F}\) are measurable then \(g\) can be also measurable. The similar result is proved for families of functions with the Baire property. In the second part of the paper questions \((1')\) and \((2')\) are considered in the class of cliquish (i.e., pointwise discontinuous) functions. In the third part of the paper the author answers those questions for finite families of functions from the first class of Baire. As a corollary he obtains the characterization of averages of Darboux functions in the first class of Baire. In particular he shows that not every \(f\in B_1\) is an average of two \(DB_1\) functions. This solves a question of A. M. Bruckner, J. G. Ceder and T. L. Pearson [Rev. Roum. Math. Pures Appl. 19, No. 3, 977-988 (19974; Zbl 0289.26005)]. (See also J. G. Ceder and T. L. Pearson’s survey article [Real Anal. Exch. 9, No. 1, 179-194 (1984; Zbl 0579.26002)]).

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
54C30 Real-valued functions in general topology
54C08 Weak and generalized continuity
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