On compositions with connected functions (Q1088798)

The author initiates a study of the following two questions: Let A and B be two classes of surjections of a given open interval I onto itself. If \(f,g\in A\) do there exist \(\alpha,\beta \in B\) such that \(\alpha \circ f=g\circ \beta\) (first equation), \(f=\alpha \circ g\circ \beta,\) resp. (second question)? He proved: If f, g are two functions of I into I such that the cardinality of f(I) is the cardinality of the continuum and g(I) is an open interval, then there exist two functions \(\alpha\), \(\beta\) of I into I such that the graphs of \(\alpha\) and \(\beta\) are connected subsets in \(I\times I\) and \(\alpha \circ f=g\circ \beta.\) If f and g are moreover surjections on I, then \(\alpha\) and \(\beta\) can be also surjections on I with connected graphs in \(I\times I.\) For any surjection f on I there exists a measurable Darboux surjection \(\alpha\) such that \(f\circ \alpha\) is measurable and Darboux. For any surjections f and g on I there exist two surjections \(\alpha\) and \(\beta\) such that \(f=\alpha \circ g\circ \beta\) iff there exists a decomposition \(\{A(y):y\in I\}\) of I into disjoint non empty sets such that for any \(y\in I\) the cardinality of the set \(\cup \{g^{-1}(z):z\in A(y)\}\) is not greater than the cardinality of \(f^{-1}(y).\) With the help of the continuum hypothesis or Martin's axiom, the author proved the existence of two functions h and g on I with connected graphs whose composition is not a function with a connected graph. Note of the reviewer: From the constructions of h and g, it is for me not evident, how \(dom h=dom g=h(I)=g(I)=I\) may hold.