Locally Lipschitz functions are generically pseudo-regular on separable Banach spaces

Abstract

For a locally Lipschitz function on a separable Banach space the set of points of Gateaux differentiability is dense but not necessarily residual. However, the set of points where the upper Dini derivative and the Clarke derivative agree is residual. It follows immediately that the set of points of intermediate differentiability is also residual and the set of points where the function is Gateaux but not strictly differentiate is of the first category. However, it is well known that the extension of differentiability theory from continuous convex to locally Lipschitz functions is fraught with difficulties. In particular,

Partial subdifferentials, derivates and Rademacher’s Theorem

“…One can now deduce very easily the following known result (see [13] for an even stronger result): Proof. The set F being proximinal (Proposition 7.4), we have (−d F )(x; 0) = 0 for all x in H \ F (Proposition 7.5).…”

Directional upper derivatives and the chain rule formula for locally Lipschitz functions on Banach spaces

“…Taking suprema in 2.15, we immediately get the result of [21] for strict upper derivative and its counterpart for other upper derivatives. (i) There is a σ-directionally porous set…”

“…Subadditivity of derived sets. Though we postpone the case of one dimensional range to the next section, we should remark that it is now easy to deduce from 2.12(iii) or from 2.15(iii) the result of [21] on generic convexity of upper derivative; indeed one can improve it by using 2.12(ii) or 2.15(ii) to a result on convexity of upper derivative with a σ-porous exceptional set. However, even this improvement is not sufficient to obtain useful information about derivative of composite functions, and we need a considerable refinement of our previous results to achieve this.…”

Typical Properties of Lipschitz Functions

“…We include a simpler proof of the following lemma due to Giles and Sciffer [4] to make our exposition self-contained. Both the statement and the proof remain valid when X is replaced by an arbitrary separable Banach space.…”