Frechet Differentiability Via Partial Frechet Differentiability

Let X-1 , ... , X-n be Banach spaces and f a real function on X = X-1 x x X-n. Let A(f) be the set of all points x is an element of X at which f is partially Frechet differentiable but is not Frechet differentiable. Our results imply that if X-1 , ... , Xn-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 : X -> Y is a Lipschitz mapping, then there exists a sigma-upper porous set A subset of X such that f is Frechet differentiable at every point x is an element of X \ A at which it is Frechet differentiable along a closed subspace of finite codimension and G & aacute;teaux differentiable. A number of related more general results are also proved.