Title: A Characterization of Continuous differentiability of Proximal Mappings of Composite Functions Ebrahim Sarabi Abstract. In this talk, we discuss a new approach toward understanding continuous differentiability of the proximal mapping of certain composite functions, which often appear in important classes of constrained and composite optimization problems. We show that such a property of the proximal mappings is equivalent to the concept of strict proto-differentiability of subgradient mappings. Using this and stability properties of generalized equations at their nondegenerate solutions, we present a characterization of smoothness of the proximal mapping for certain composite functions. Extensions to the prox-regular sets and functions will be discussed at the end.