Can one recognize a function from its graph? (Q6060932)

In the present paper the authors want to answer the following two dual questions. \begin{itemize} \item Given a class of functions \(f\colon X\to Y\), where \(X\) and \(Y\) are metric spaces, can we find a property of their graphs as subsets of \(X\times Y\) which exactly characterizes functions from this class? \item Given a property of an appropriate subset of \(X\times Y\), can we describe the exact class of functions \(f\colon X\to Y\) such that their graphs have this property? \end{itemize} With a number of lemmas and theorems, the following classes of functions are characterized in this way: \begin{itemize} \item functions with closed graphs; \item functions with connected graphs; \item functions with pathwise connected graphs; \item regular Darboux functions. \end{itemize} In the second part of the paper the authors analyze the following problem: \begin{center} If the graphs of two functions \(f\colon X\to Y\) and \(g\colon Y\to Z\) have a certain topological property, does the graph of their composition \(g\circ f\colon X\to Z\) have the same property? \end{center} They give some answers on this problem concerning the classes of functions with closed graphs, functions with connected graphs and functions which have closed fibres. Throughout the paper there are many examples and counterexamples illustrating the relationships between the discussed classes of functions.