Derivatives and Lebesgue points via homeomorphic changes of scale

Author:
Don L. Hancock

Journal:
Trans. Amer. Math. Soc. **267** (1981), 197-218

MSC:
Primary 26A24

DOI:
https://doi.org/10.1090/S0002-9947-1981-0621982-1

MathSciNet review:
621982

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Abstract: Let be a closed interval, and suppose , , and denote, respectively, the class of homeomorphisms of onto itself, the class of homeomorphisms of the line onto itself, and the class of real functions on for which each point is a Lebesgue point. Maximoff proved that is exactly the class of Darboux Baire functions, where . The present paper is devoted primarily to a study of . The characterizations of this class which are obtained show that a function is a member of if and only if, in addition to the obvious requirement of approximate continuity, it satisfies certain growth and density-like conditions. In particular, any approximately continuous function with countably many non-Lebesgue points belongs to . It is also established that is a uniformly closed algebra properly containing the smallest algebra generated from , and a characterization of the latter algebra is provided.

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DOI:
https://doi.org/10.1090/S0002-9947-1981-0621982-1

Article copyright:
© Copyright 1981
American Mathematical Society