Discontinuously basic sets and the 13th problem of Hilbert

A subset $M\subset \mathbb{R}^3$ is called a discontinuously basic subset, if for any function $f \colon M \to \mathbb{R}$ there exist such functions $f_1; f_2; f_3 \colon \mathbb{R} \to \mathbb{R}$ that $f(x_1, x_2, x_3) = f_1(x_1) + f_2(x_2) + f_3(x_3)$ for each point $(x_1, x_2, x_3)\in M$. We will prove a criterion for a discontinuous basic subset for some specific subsets in terms of some graph properties. We will also introduce several constructions for minimal discontinuous non-basic subsets.