Derivatives and Lebesgue points via homeomorphic changes of scale

Hancock, Don L.

doi:10.1090/S0002-9947-1981-0621982-1

Derivatives and Lebesgue points via homeomorphic changes of scale
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by Don L. Hancock PDF

Trans. Amer. Math. Soc. 267 (1981), 197-218 Request permission

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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