Functions with a concave modulus of continuity

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 $> \alpha$.