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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 $f|C$ 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 $f|C$ 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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Keywords: Darboux, Baire 1, <IMG WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${L_p}$">-smooth, <IMG WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img17.gif" ALT="${L_p}$">-derivative, Denjoy-Clarkson Property
Article copyright: © Copyright 1976 American Mathematical Society