Density of Robustly Quasiconvex Functions in Quasiconvex Functions

In this paper we prove that the set of robustly quasiconvex functions is dense in the set of quasiconvex functions that obtain global minimum values on a closed bounded interval of $$X=\mathbb {R}$$ and a real-valued function defined on a convex set in a normed linear space X is quasiconvex iff the robustness radius of its epigraph is non-negative. In addition, a function is robustly quasiconvex if and only if its epigraph in $$X\times \mathbb {R}$$ is robustly horizontally convex. A preservation property of convexity for robustly horizontally convex sets is established.