Discontinuous Strongly Quasiconvex Functions

TL;DR

The paper addresses whether real-valued strongly quasiconvex functions on $\mathbb{R}^n$ must be continuous, and provides explicit counterexamples demonstrating discontinuities. It develops constructions for univariate and multivariate SQC functions that are discontinuous, including cases with infinitely many discontinuities on the boundary and in the interior of the effective domain. The results also explore the (non)occurrence of lower and upper semicontinuity across infinitely many points and discuss implications for coercivity and optimization theory. Overall, the work delineates the limits of continuity for SQC functions and reveals rich boundary behaviors within finite-dimensional settings.

Abstract

A fundamental open question asking whether all real-valued strongly quasiconvex functions defined on $\mathbb R^n$ are necessarily continuous, akin to their convex counterparts, is answered in detail in this paper. Among other things, we show that such functions can have infinitely many points of discontinuity. The failure of lower semicontinuity together with the lack of upper semicontinuity at infinitely many points of certain real-valued strongly quasiconvex functions are also shown.