Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 26A15, 26A24, 30C15

Retrieve articles in all journals with MSC: 26A15, 26A24, 30C15


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1203978-9
PII: S 0002-9939(1994)1203978-9
Keywords: Generalized continuity, generalized Riemann derivative of order zero
Article copyright: © Copyright 1994 American Mathematical Society