An absorption property for the composition of functions (Q1308838)

The authors have introduced the following notion: if \(F\) is a class of functions transforming \(X\) into itself such that at least one function in \(F\) is a surjection, then we say that \(F\) has the right absorption property (RAP) if and only if the following condition holds: if \(g: X\to X\) is such that \(g\circ f\in F\) for some surjection \(f\in F\), then \(g\in F\). The authors have shown that the class of all analytic functions from the complex plane into itself has RAP and the class of analytic functions from the real line into itself has not. Also the class of continuous functions from \(\mathbb{R}^ n\) to \(\mathbb{R}^ n\) has RAP if and only if \(n=1\) and the class of \(k\) times differentiable functions from \(\mathbb{R}^ n\) to \(\mathbb{R}^ n\) does not have RAP for any \(k\) and \(n\). Neither the class of measurable functions, nor the class of functions having the Baire property (from \(\mathbb{R}\) to \(\mathbb{R}\)) has RAP and this fact follows from the general criterion (Theorem 2.4).