Discontinuous Strongly Quasiconvex Functions

TL;DR

This paper investigates the continuity properties of strongly quasiconvex functions on real space, revealing that such functions can be discontinuous at infinitely many points and may lack semicontinuity.

Contribution

It provides a detailed answer to whether all strongly quasiconvex functions are necessarily continuous, showing they can be discontinuous and lack semicontinuity.

Findings

Strongly quasiconvex functions can have infinitely many discontinuities

Such functions can lack lower semicontinuity at many points

They can also lack upper semicontinuity at infinitely many points

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.