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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
MathSciNet review: 621982
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Abstract: Let $ I$ be a closed interval, and suppose $ \mathcal{K}$, $ \mathcal{H}$, and $ \Lambda $ denote, respectively, the class of homeomorphisms of $ I$ onto itself, the class of homeomorphisms of the line onto itself, and the class of real functions on $ I$ for which each point is a Lebesgue point. Maximoff proved that $ \Lambda \circ \mathcal{K}$ is exactly the class of Darboux Baire $ 1$ functions, where $ \Lambda \circ \mathcal{K} = \{ f \circ k:f \in \Lambda ,k \in \mathcal{K}\} $. The present paper is devoted primarily to a study of $ \mathcal{H} \circ \Lambda = \{ h \circ f:f \in \Lambda ,h \in \mathcal{H}\} $. The characterizations of this class which are obtained show that a function is a member of $ \mathcal{H} \circ \Lambda $ 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 $ \mathcal{H} \circ \Lambda $. It is also established that $ \mathcal{H} \circ \Lambda $ is a uniformly closed algebra properly containing the smallest algebra generated from $ \Lambda $, and a characterization of the latter algebra is provided.

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Article copyright: © Copyright 1981 American Mathematical Society