We study continuity and differentiability properties for a reformulation of a finite-dimensional quasi-variational inequality (QVI) problem using a regularized gap function approach. Since the regularized gap function is nonsmooth in general, we take a closer look at its nondifferentiability points and show, in particular, that under mild assumptions all locally minimal points of the reformulation are, in fact, differentiability points. The results are specialized to generalized Nash equilibrium problems, and consequences for numerical methods are discussed.