Specular differentiation in normed vector spaces: Quasi-Mean Value and Quasi-Fermat Theorems

Abstract

This paper introduces specular differentiation, which generalizes Gâteaux and Fréchet differentiation in normed vector spaces. We investigate its fundamental theoretical properties and establish weak forms of the Mean Value Theorem and Fermat's Theorem in the specular sense. Finally, we identify a distinguished element of the Fréchet subdifferential of a convex function through specular differentiation.

24 pages. This paper is part of a split of the previous preprint arXiv:2601.10950v1, and addresses its theoretical foundations