Baire functions and their restrictions to special sets.

*(English)*Zbl 0839.26003Real functions on the line and their restrictions to general nonvoid, countable nonvoid, perfect or \(\sigma\)-perfect sets are considered. It is asked, what can be said about such an \(f\) if all restrictions to sets of a given type have a point or a portion of continuity, quasi-continuity, or cliquishness. As concerns cliquishness, the question is answered completely – in any case those \(f\) are precisely all Baire one functions. For the two other properties several results are derived. Together with the result of T. Natkaniec [Math. Slovaca 43, No. 4, 455-457 (1993; preceding review)] and a remark by the reviewer in Real. Anal. Exch. 18, No. 2, 385-399 (1993)] the following is obtained. In all but two cases the class of such \(f\) is either the class of Baire one functions or the class of Baire one star functions. The only cases where no characterization is given are the classes of functions for which each restriction to a \(\sigma\)-perfect set has a point of continuity or quasi-continuity. However, in case \(f\) is Darboux, additional results follow.

Reviewer: B.Kirchheim (Freiburg)

##### 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 |

##### Keywords:

Darboux functions; continuity; quasi-continuity; cliquishness; Baire one functions; Baire one star functions
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