Characterizations and generalizations of continuity
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- by J. M. Ash, J. Cohen, C. Freiling, L. Gluck, E. Rieders and G. Wang PDF
- Proc. Amer. Math. Soc. 121 (1994), 833-842 Request permission
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
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Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 121 (1994), 833-842
- MSC: Primary 26A15; Secondary 26A24, 30C15
- DOI: https://doi.org/10.1090/S0002-9939-1994-1203978-9
- MathSciNet review: 1203978