Two examples concerning derivatives and \(M_ 3\)-sets (Q1071879)

There are sets \(\emptyset \neq C_ 1,C_ 2\subset [0,1]\) which can be separated by a derivative, but cannot be separated by any function of the form F(x,f(x)), where f is a bounded derivative and \(F: R^ 2\to R\) is a continuous function. There is a set in the \(M_ 3\)-Zahorski class which cannot be written as \(f^{-1}(G)\), where f is a derivative and G is an open set.