Baire functions and their restrictions to special sets. (English) Zbl 0839.26003

Real 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.


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


Zbl 0839.26002
Full Text: EuDML


[1] BLEDSOE W.: Neighborly functions. Proc. Amer. Math. Soc. 3 (1952), 114-115. · Zbl 0046.40301
[2] BRUCKNER A. M.: Differentiation of Real Functions. Lecture Notes in Math. 659, Springer Verlag, Berlin-Heidelberg-New York, 1978. · Zbl 0382.26002
[3] CSASZAR A., LACZKOVICH M.: Discrete and equal convergence. Studia Sci. Math. Hungar. 10 (1975), 463-472. · Zbl 0405.26006
[4] GRANDE Z.: The Darboux property in some families of Baire 1 functions. Tatra Mountains Math. Publ 2 (1993), 7-14. · Zbl 0788.26002
[5] GRANDE Z.: Quelques remarques sur les familles de fonctions de premi√®re classe. Fund. Math. 84 (1974), 87-91. · Zbl 0279.26004
[6] KEMPISTY S.: Sur les fonctions quasicontinues. Fund. Math. 19 (1932), 184-197. · Zbl 0005.19802
[7] PEEK D. E.: Baire functions and their restrictions to special sets. Proc. Amer. Math. Soc. 30 (1971), 303-307. · Zbl 0229.26006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.