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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Baire* $ 1$, Darboux functions


Author: Richard J. O’Malley
Journal: Proc. Amer. Math. Soc. 60 (1976), 187-192
MSC: Primary 26A21; Secondary 26A24
MathSciNet review: 0417352
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Abstract: It is well known that a function $ f:[0,1] \to R$ is Baire 1 if and only if in any closed set C there is a point $ {x_0}$ at which the restricted function $ f\vert C$ is continuous. Functions will be called Baire$ ^\ast$ 1 if they satisfy the following stronger property: For every closed set C there is an open interval (a, b) with $ (a,b) \cap C \ne \emptyset $ such that $ f\vert C$ is continuous on (a, b). Functions which are both Baire$ ^\ast$ 1 and Darboux are examined. It is known that approximately derivable functions are Baire$ ^\ast$ 1. Among other things it is shown here that $ {L_p}$-smooth functions are Baire$ ^\ast$ 1. A new result about the $ {L_p}$-differentiability of $ {L_p}$-smooth, Darboux functions is shown to follow immediately from the main properties of Baire$ ^\ast$ 1, Darboux functions.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0417352-5
PII: S 0002-9939(1976)0417352-5
Keywords: Darboux, Baire 1, $ {L_p}$-smooth, $ {L_p}$-derivative, Denjoy-Clarkson Property
Article copyright: © Copyright 1976 American Mathematical Society