Closure of Darboux graphs (Q1803962)

The purpose of the paper is to find a suitable class of functions with such a property that the graph of any Darboux function \(f:I\to I\) would have the same closure in \(I^ 2\) as some member of this class. In 1974 \textit{H. Miller} showed the fact that the class of connectivity functions can be considered as a suitable example of such a class. The main result of the paper gives an answer to the question of \textit{K. Kellum} whether the result of Miller can be generalized in a certain way. More precisely, it is shown here that the connectivity function, the graph of which has the same closure as the graph of the given Darboux function, can be extended to a connectivity function from \(I^ 2\) to \(I\). Moreover an example is given to show that Miller's result cannot be generalized to Darboux functions from \(I^ 2\) to \(I^ 2\).