Functions with a concave modulus of continuity
Abstract: In , C. Goffman proved that, if is a modulus of continuity, then the set of all functions f in such that (m denotes Lebesgue measure) for all g in , the set of all functions in having as a modulus of continuity, is residual in . In the present article, we prove that, if is a concave modulus of continuity and , then the set of all functions f in such that for all g in is residual in . Using this result, we show that, if , then there are functions in which satisfy a Hölder condition of exponent such that for all g in which satisfy a Hölder condition of exponent .
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 26A15
Retrieve articles in all journals with MSC: 26A15
Keywords: Modulus of continuity, Hölder condition, concave modulus of continuity
Article copyright: © Copyright 1974 American Mathematical Society