Note on functions satisfying the integral Hölder condition

summary:Given a modulus of continuity $\omega$ and $q \in[1, \infty[ $ then $H_q^\omega$ denotes the space of all functions $f$ with the period $1$ on $\R$ that are locally integrable in power $q$ and whose integral modulus of continuity of power $q$ (see(1)) is majorized by a multiple of $ \omega$. The moduli of continuity $ \omega$ are characterized for which $H_q^\omega$ contains "many" functions with infinite "essential" variation on an interval of length $1$