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Functions with a concave modulus of continuity


Author: H. E. White
Journal: Proc. Amer. Math. Soc. 42 (1974), 104-112
MSC: Primary 26A15
DOI: https://doi.org/10.1090/S0002-9939-1974-0330366-7
MathSciNet review: 0330366
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Abstract: In [1], C. Goffman proved that, if $ \sigma $ is a modulus of continuity, then the set of all functions f in $ C[0,1]$ such that $ m(\{ x:f(x) = g(x)\} ) = 0$ (m denotes Lebesgue measure) for all g in $ C(\sigma )$, the set of all functions in $ C[0,1]$ having $ \sigma $ as a modulus of continuity, is residual in $ C[0,1]$. In the present article, we prove that, if $ \sigma $ is a concave modulus of continuity and $ 0 < K < {24^{ - 1}}$, then the set of all functions f in $ C(\sigma )$ such that $ m(\{ x:f(x) = g(x)\} ) = 0$ for all g in $ C(K\sigma )$ is residual in $ C(\sigma )$. Using this result, we show that, if $ 0 < \alpha < 1$, then there are functions in $ C[0,1]$ which satisfy a Hölder condition of exponent $ \alpha $ such that $ m(\{ x:f(x) = g(x)\} ) = 0$ for all g in $ C[0,1]$ which satisfy a Hölder condition of exponent $ > \alpha $.


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

  • [1] C. Goffman, Approximation of non-parametric surfaces of finite area, J. Math. Mech. 12 (1963), 737-745. MR 27 #3782. MR 0153821 (27:3782)
  • [2] W. S. Loud, Functions with prescribed Lipschitz conditions, Proc. Amer. Math. Soc. 2 (1951), 358-360. MR 13, 218. MR 0043182 (13:218j)

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DOI: https://doi.org/10.1090/S0002-9939-1974-0330366-7
Keywords: Modulus of continuity, Hölder condition, concave modulus of continuity
Article copyright: © Copyright 1974 American Mathematical Society

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