Every real function is the sum of two extendable connectivity functions (Q1912734)

A function \(F: \mathbb{R}^2\to \mathbb{R}\) is a \textit{connectivity function} if the graph of its restriction \(F|C\) to any connected \(C\subset \mathbb{R}^2\) is connected in \(\mathbb{R}^3\). A function \(f: \mathbb{R}\to \mathbb{R}\) is an \textit{extendable function} if there exists a connectivity function \(F: \mathbb{R}^2\to \mathbb{R}\) with \(f(x)= F(x, 0)\) for every \(x\in \mathbb{R}\). The author constructs an extendable function, whose graph is dense in \(R\times R\) and uses this example to prove that every real function from \(\mathbb{R}\) to \(\mathbb{R}\) can be written as the sum of two extendable functions.