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Keywords:
Fréchet differentiability; partial Fréchet differentiability; first category set; Asplund space; $\sigma$-porous set
Fréchet differentiability; partial Fréchet differentiability; first category set; Asplund space; $\sigma$-porous set
Summary:
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.
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