Derivatives on countable dense subsets (Q1086370)

The author proves the following result: Let \((A,B,C,D)\) and \((A_ 0,B_ 0,C_ 0,D_ 0)\) be quadruples of mutually disjoint countable dense subsets of \(\mathbb R\) \((\mathbb R\) -- the real line). Then there exists a continuously differentiable function \(g: \mathbb R\to \mathbb R\) such that \(g(A)=A_ 0\), \(g(B)=B_ 0\), \(g(C)=C_ 0\), \(g(D)=D_ 0\), and \(g'\geq 1\) on \(\mathbb R\). This result gives a possibility to construct a variety of types of pathological functions.