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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Baire* $1$, Darboux functions
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by Richard J. O’Malley PDF
Proc. Amer. Math. Soc. 60 (1976), 187-192 Request permission

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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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 60 (1976), 187-192
  • MSC: Primary 26A21; Secondary 26A24
  • DOI: https://doi.org/10.1090/S0002-9939-1976-0417352-5
  • MathSciNet review: 0417352