Some remarks on the class of continuous (semi-) strictly quasiconvex functions

We introduce the notion of variational (semi-) strict quasimonotonicity for a multivalued operator T : X double right arrow X* relative to a nonempty subset A of X which is not necessarily included in the domain of T. We use this notion to characterize the subdifferentials of continuous (semi-) strictly quasiconvex functions. The proposed definition is a relaxation of the standard definition of (semi-) strict quasimonotonicity, the latter being appropriate only for operators with nonempty values. Thus, the derived results are extensions to the continuous case of the corresponding results for locally Lipschitz functions.