On Problem 57 of the Scottish book (Q2322493)

The paper provides a new proof of a classical result in real analysis: Given two functions \(w\) and \(\varphi\) such that both \(w(h)\) and \(\varphi(h)\) decrease to \(0\) as \(\vert h\vert \to 0\) and that \(\lim_{h \to 0} h{-1} w(h) = \lim_{h \to 0} w(h) / \varphi(h) = \infty\), there exists a function \(f\) with the properties (a) \(\vert f(x+h) - f(x)\vert < w(h)\) and (b) \(\limsup_{h \to 0} \vert f(x+h) - f(x)\vert / \vert \varphi(h)\vert = \infty\) holding for all \(x\) and \(h \ne 0\). In contrast to the already known proof, this new one is based on techniques from Baire category theory. A particular implication of this result is that there exist nowhere differentiable functions that satisfy Lipschitz conditions of order \(\alpha \in (0,1)\).