When is f(x_1, x_2, . . ., x_n)=u_1(x_1)+u_2(x_2)+ . . . +u_n(x_n)?

We discuss subsets $S$ of ${\mathbb{R}}^n$ such that every real valued function $f$ on $S$ is of the form {equation*} f(x_1,x_2,…,x_n) = u_1(x_1) + u_2(x_2) + … + u_n(x_n), {equation*} and the related concepts and situations in analysis.