Abstract: Let X_1, \dots, X_n be Banach spaces and f a real function on X=X_1 \times\dots \times X_n. Let A_f be the set of all points x \in X at which f is partially Fréchet differentiable but is not Fréchet differentiable. Our results imply that if X_1, \dots, X_{n-1} are Asplund spaces and f is continuous (respectively Lipschitz) on X, then A_f is a first category set (respectively a \sigma-upper porous set). We also prove that if X, Y are separable Banach spaces and f\colon X \to Y is a Lipschitz mapping, then there exists a \sigma-upper porous set A \subset X such that f is Fréchet differentiable at every point x \in X \setminus A at which it is Fréchet differentiable along a closed subspace of finite codimension and Gâteaux differentiable. A number of related more general results are also proved.
Keywords: Fréchet differentiability; partial Fréchet differentiability; first category set; Asplund space; \sigma-porous set
DOI: DOI 10.14712/1213-7243.2023.025
AMS Subject Classification: 46G05 46T20