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The Darboux property in some families of Baire 1 functions. (English) Zbl 0788.26002
Summary: Denote by \(D\) the family of Darboux functions, by \(P\) the family of Peek functions [D. Peek, Proc. Am. Math. Soc. 30, 303-307 (1971; Zbl 0229.26006)] pointwise discontinuous on each union of sequences of perfect sets, by \(G_ 1\) the family of functions pointwise discontinuous on each non-empty set, and by \(Q\) the family of quasicontinuous functions. I investigate the addition and the multiplication in \(P\) and \(DP\). Moreover, I show thatat \(DP\subset Q\), and that \(f\in G_ 1\) iff for every perfect set \(E\) there is an open interval \(I\) such that \(I\cap E\neq\emptyset\) and \(f| (I\cap E)\) is continuous.

26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
26A21 Classification of real functions; Baire classification of sets and functions
26A03 Foundations: limits and generalizations, elementary topology of the line