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Proceedings of the American Mathematical Society

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

 

 

On functions with summable approximate Peano derivative


Author: Cheng Ming Lee
Journal: Proc. Amer. Math. Soc. 57 (1976), 53-57
DOI: https://doi.org/10.1090/S0002-9939-1976-0399379-5
MathSciNet review: 0399379
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Abstract: Let $ n$ be a positive integer and $ F$ a function defined on a closed interval $ I$. For $ x$ in $ I$, let the $ n$th approximate Peano derivative of $ F$ at $ x$, if it exists, be denoted as $ {F_{(n)}}(x)$. For $ n = 1$, the existence of $ {F_{(n - 1)}}(x)$ will simply mean that the function $ {F_{(0)}}( \equiv F)$ is approximately continuous at $ x$. Then the following theorem is proved, noting that the phrase ``for nearly all $ x$ in $ I$'' means ``for all $ x$ in $ I$ except perhaps for those points in a countable subset of $ I$", Theorem $ {{\mathbf{A}}_n}$. Let $ {F_{(n - 1)}}(x)$ exist finitely for all $ x$ in $ I$. If $ {F_{(n)}}(x)$ exists finitely for nearly all $ x$ in $ I$ and is summable on $ I$, then $ {F_{(n - 1)}}$ is absolutely continuous in $ I$.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0399379-5
Keywords: Approximate continuity, approximate Peano derivative, absolute continuity, generalized absolute continuity, monotone increasing, summable functions
Article copyright: © Copyright 1976 American Mathematical Society