Continuity of Darboux functions with nice finite iterations (Q2464625)

The main results: 1) If \(g\) is a nowhere constant continuous function and \(f\) is a Darboux function such that there is \(m\) and for each \(x\) there is a positive integer \(n_x \leq m\) with \(f^{n_x}(x) = g(x)\), then \(f\) is continuous. 2) If for a continuous function \(g\) there is an open interval \(I\) with \(g/I = k \in I\) then there is a Darboux function \(f\) such that for each \(x\) there is an integer \(n_x\leq 2\) with \(f^{n_x} = g(x)\). Moreover, two examples are given showing that the assumption that \(k\in I\) is important.