Differential properties of crinkly functions (Q1078342)

A continuous function \(f: D\to R\), where D is a nondegenerate closed interval in R, is said to be crinkly iff for any nondegenerate closed subinterval I of D and for any \(c\in (\min f(I),\max f(I))\) the set \(I\cap f^{-1}\{c\})\) is uncountable. The author proved the following theorem on crinkly functions and their Dini derivatives \(f^+\), \(f^-\), \(f_+\) and \(f_-:\) For any closed interval \(<a,b>,\quad a<b,\) for any function \(H: <0,b-a>\to (0,\infty)\) for which \(\lim_{d\to 0+}H(d)=\infty\) and \(\lim_{d\to 0+}(dH(d))=0\) and for any four disjoint Lebesgue measurable sets \(E_ 1,...,E_ 4\), there exists a crinkly function \(f: <a,b>\to R\) having the following properties: 1. \(f^+=f^-=\infty\) and \(f_+=f_-=-\infty\) holds a.e. on \(E_ 1,\) 2. \(f^+=\infty\), \(f_-=-\infty\) and \(f_+=f^-=0\) holds a.e. on \(E_ 2,\) 3. \(f_+=-\infty\), \(f^-=\infty\) and \(f_-=f^+=0\) holds a.e. on \(E_ 3,\) 4. \(f^+=f^-=f_+=f_-=0\) holds a.e. on \(E_ 4\) and \(\omega\) (d,f)\(\leq dH(d)\) for any \(d\in <0,b-a>\), where \(\omega\) (d,f) is the modulus of continuity of f for d.