Continuity characterisations of differentiability of locally Lipschitz functions

Abstract

Recently David Preiss contributed a remarkable theorem about the differentiability of locally Lipschitz functions on Banach spaces which have an equivalent norm differentiable away from the origin. Using his result in conjunction with Frank Clarke's non-smooth analysis for locally Lipschitz functions, continuity characterisations of differentiability can be obtained which generalise those for convex functions on Banach spaces. This result gives added information about differentiability properties of distance functions.For a continuous convex function <f> on an open convex subset A of a real normed linear space X where <j> is Gateaux differentiable on a dense subset D of A, it is known that <j> is Gateaux (Frechet) differentiable at x £ A if and only if <j)'(x n ) is weak * (norm) convergent for all x n £ D converging to x.In [5, Corollary 2, p.64] it was shown that a similar result holds for a distance function d on a normed linear space X with uniformly Gateaux (uniformly Frechet) differentiable norm.Here we prove that such a continuity characterisation of differentiability applies with locally Lipschitz functions on a real Banach space with equivalent norm Gateaux differentiable away from the origin. This result in turn contributes further to our knowledge of differentiability properties of distance functions. But it is also of use in non-smooth optimisation relying as it does on the Clarke generalised subdifferential which is not in itself a good indicator of differentiability.A real function ^ on an open subset A of a normed linear space X is said to be locally Lipachitz on A if for each x £ A there exists a K > 0 and a S > 0 such thatSuch a function <j> is said to be Gateaux differentiable at x £ A if there exists a continuous linear functional fi(x) on X where, given e > 0 and ||i/|| = 1 there exists a 6(e, x, y) > 0 such that (x + ty) -4>{x) , < e when 0 < \t\ < 6.

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

“…But as both V' + ( :c )(l/) anc^ ^"(^Jtv) a r e continuous in y we conclude that for every xeD+ V»+(x)(y) = i>°(x)(y) for all y € X. [5] Locally …”

“…tl>'{x)(y) -ip°{x)(y) for all y £ X then we say that ip is strictly differentiable at z. If V 1 is Gateaux differentiable on a dense subset D ol A and is strictly differentiable at x £ A then the derivatives V"'( z ) a r e weak * convergent to V>'(x) for z £ Z> as z approaches x, [5]. We say that V 1 is regular at x £ A if ip has a right hand derivative at x and V'+( a; )(2/) = V >°(x )(y) f°r all y £ -X^ and is pseudo-regular at z £ A if V l+ ( a; )(y) = V |0 ( a: )(y) f°r all y £ X , [2].…”