An extreme function which is nonnegative and discontinuous everywhere

Abstract

We consider Gomory and Johnson's infinite group model with a single row. Valid inequalities for this model are expressed by valid functions and it has been recently shown that any valid function is dominated by some nonnegative valid function, modulo the affine hull of the model. Within the set of nonnegative valid functions, extreme functions are the ones that cannot be expressed as convex combinations of two distinct valid functions. In this paper we construct an extreme function $\pi:\mathbb{R} \to [0,1]$ whose graph is dense in $\mathbb{R} \times [0,1]$. Therefore, $\pi$ is discontinuous everywhere.