Specular differentiation in normed vector spaces

This paper develops specular differentiation for mappings between real normed vector spaces, extending the classical notions of Gâteaux and Fréchet differentiability. These notions apply to a broad class of nonsmooth functions, including functions for which certain other generalized derivatives do not exist. We investigate their fundamental theoretical properties and establish a Quasi-Mean Value Theorem for real-valued functions. Finally, we prove that the specular Gâteaux derivative of a semi-convex functional, whenever it exists at an interior point of its effective domain, determines a global quadratic lower bound and belongs to its Fréchet subdifferential.