Baire* , 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 is Baire 1 if and only if in any closed set *C* there is a point at which the restricted function is continuous. Functions will be called Baire 1 if they satisfy the following stronger property: For every closed set *C* there is an open interval (*a, b*) with such that is continuous on (*a, b*). Functions which are both Baire 1 and Darboux are examined. It is known that approximately derivable functions are Baire 1. Among other things it is shown here that -smooth functions are Baire 1. A new result about the -differentiability of -smooth, Darboux functions is shown to follow immediately from the main properties of Baire 1, Darboux functions.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0417352-5

Keywords:
Darboux,
Baire 1,
-smooth,
-derivative,
Denjoy-Clarkson Property

Article copyright:
© Copyright 1976
American Mathematical Society