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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Characterizations and generalizations of continuity

Authors: J. M. Ash, J. Cohen, C. Freiling, L. Gluck, E. Rieders and G. Wang
Journal: Proc. Amer. Math. Soc. 121 (1994), 833-842
MSC: Primary 26A15; Secondary 26A24, 30C15
MathSciNet review: 1203978
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Abstract: The condition $ f(x + 2h) - 2f(x + h) + f(x) = o(1)$ (as $ h \to 0$) at each x is equivalent to continuity for measurable functions. But there is a discontinuous function satisfying $ 2f(x + 2h) - f(x + h) - f(x) = o(1)$ at each x. The question of which generalized Riemann derivatives of order 0 characterize continuity is studied. In particular, a measurable function satisfying $ \sum\nolimits_{i = 1}^n {{\alpha _i}f(x + {\beta _i}h) \equiv 0} $ must be a polynomial. On the other hand, for any Riemann derivative of order 0 and any $ p \in [1,\infty ]$, generalized $ {L^p}$ continuity is equivalent to $ {L^p}$ continuity almost everywhere.

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PII: S 0002-9939(1994)1203978-9
Keywords: Generalized continuity, generalized Riemann derivative of order zero
Article copyright: © Copyright 1994 American Mathematical Society