Compositions of Darboux-like functions (Q1978855)

In the paper three notions concerning real functions are studied: a function \(f:\mathbf R\rightarrow \mathbf R\) is almost continuous if, given a closed set \(K\subset \mathbf R^2\) such that graph\((f)\cap K=\emptyset \), there exists a continuous function \(g:\mathbf R\rightarrow \mathbf R\) such that graph\((g)\cap K=\emptyset \). A function \(f\) is Darboux if \(f(C)\) is connected whenever \(C\) is connected; \(f\) is a connectivity function if graph\((f |C)\) is connected whenever \(C\) is connected. It is shown that there exists a connectivity function which cannot be written as the composition of finitely many almost continuous functions. This result gives a negative answer to the question if every Darboux function from \textbf{R} to \textbf{R} is the composition of two almost continuous functions. Moreover, it is shown what can be said about Darboux function \(g\) and \(f\) when their composition \(g\circ f\) is continuous. At the end the author recalls the following open question of \textit{J. G. Ceder} [Real Anal. Exch. 11, 380-389 (1986; Zbl 0613.26012)]: Is every real Darboux function the composition of two (finitely many) connectivity functions?