Uniform approximation by polynomials in two functions (Q581699)

The main aim of this very interesting paper is the proof of the following Theorem. Let f: \(C\to C\) be proper (that is f is a continuous function for which the pre-image of each compact set is compact). Suppose \(E\subset C\) is closed, int E\(=\emptyset\), and \(C\setminus E\) has no bounded components. Suppose \(0<r\in {\mathbb{Z}}\), and for each \(a\in C\setminus E\) the equality \(f(z)=f(a)\) has exactly r solutions \(a_ 1,a_ 2,...,a_ r\), \(a_ i\neq a_ j\), \(a_ i\in C\setminus E\), \(i=1,2,...,r\). Suppose that on \(C\setminus E\), f is \(C^ 1\), nonsingular, and locally direction-reversing. Then C[z,f] is dense in \(C^ 0(C,C)\), that is, for each compact set \(X\subset C\), C[z,f] is dense in C(X).