A subspace of a Hölder space, which consists only of nonsmooth functions. (Q1889539)

Let \(X\) be the Banach space \(H(\omega)\) of functions on \([0,1]\) having an \(\omega\)-estimate on the modulus of continuity, where \(\omega : [0,1] \to [0, +\infty)\) is concave, \(\omega(0)=0\) and \(\lim_{t \to 0} \frac{\omega(t)}{t}=\infty\). A particular case is the Hölder function space \(X=H_\alpha\). The main results are as follows. \(X\) contains an infinite-dimensional closed subspace consisting of functions having at each point as bad order of smoothness as it is possible in principle for elements of \(X\). The subset of functions \(f \in X\) having ``good'' smoothness on a subset of positive measure is of the first Baire category.