Clarke导数的连续性与函数的严格可微性

STRICT DIFFERENTIABILITY OF NONSMOOTH FUNCTIONS

引进Lipschitz正则条件,证明了Banach空间上Lipschitz正则函数在某点严格可微当且仅当沿每个方向的Clarke导数在该点连续;在可分空间中Lipschitz正则函数的严格可微点集是第二纲的,因而处处稠密,并得到Clarke梯度用严格导数的极限表示。

This paper proposes Lipschitz reguvarty,proves that the fun-ctions with Lipschitz regularty are strictly differentiable at x in Banachspace X if and only if Clarke's derivatives are continuous at x for everyv∈X; On Separable space,the set of the points where the functions withLipschitz regularty are strictly differentiable is second category,and Clar-kes' generalized gradient is exprssed with the limit of strict derivatives.