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Proceedings of the American Mathematical Society

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

 

 

Lebesgue sets and insertion of a continuous function


Author: Ernest P. Lane
Journal: Proc. Amer. Math. Soc. 87 (1983), 539-542
MSC: Primary 54C05; Secondary 54C30
DOI: https://doi.org/10.1090/S0002-9939-1983-0684654-0
MathSciNet review: 684654
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Abstract: Necessary and sufficient conditions in terms of Lebesgue sets are presented for the following two insertion properties for real-valued functions defined on a topological space: (1) $ g \leqslant f$ there is a continuous function $ h$ such that $ g \leqslant h \leqslant f$, and for each $ x$ for which $ g(x) < f(x)$ then $ g(x) < h(x) < f(x)$. (2) $ g < f$ there is a continuous function $ h$ such that $ g < h < f$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0684654-0
Keywords: Insertion of continuous functions, Lebesgue set, completely separated
Article copyright: © Copyright 1983 American Mathematical Society