## 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.

### MSC:

 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

Zbl 0229.26006