Transformations of functions (Q1073199)

Consider the following problem: Let A be a class of real-valued functions defined on the line. What is the class \({\mathcal H}{\mathcal A}=\{h\circ f;\quad h\quad is\quad a\quad \hom eomorphism\quad and\quad f\in {\mathcal A}\}?\) The author gives a survey of known results for such classes \({\mathcal A}\), like the class of functions of bounded variation, absolutely continuous or differentiable functions, etc. Then he shows the difficulties connected with the characterization of other classes and proves certain auxiliary results. The paper contains interesting ideas and also some open problems.