On \(G\)-semidifferentiable functions in Euclidean spaces
We give a criterion for a function \(f: \mathbb{R}^n\to \mathbb{R}\) to be upper \(G\)-semidifferentiable in the sense of \textit{F. Gianessi} [ibid. 60, No. 2, 191-241 (1989; Zbl 0662.90071)] at a point \(\overline x\in \mathbb{R}^n\). Using this result, we describe upper \(G\)-semiderivatives when \(G\) is, for instance, one of the following basic classes of homogeneous functions: the set of all continuous positively homogeneous functions, the set of differences of two sublinear functions, and the set of sublinear functions. As a result, connections between upper \(G\)- semidifferentiability and the concepts of differentiability are obtained.
scientific article; zbMATH DE number 790572