Clarke critical values of subanalytic Lipschitz continuous functions

We prove that any subanalytic locally Lipschitz function has the Sard property. Such functions are typically nonsmooth and their lack of regularity necessitates the choice of some generalized notion of gradient and of critical point. In our framework these notions are defined in terms of the Clarke and of the convex-stable subdifferentials. The main result of this note asserts that for any subanalytic locally Lipschitz function the set of its Clarke critical values is locally finite. The proof relies on Pawlucki's extension of the Puiseuxlemma. In the last section we give an example of a continuous subanalytic function which is not constant on a segment of "broadly critical" points, that is, points for which we can find arbitrarily short convex combinations of gradients at nearby points.

The microscope : an introduction to microscopic methods and to histology / by Simon Henry Gage.

11th ed

The microscope and histology for the use of laboratory students in the anatomical department of Cornell University. by Simon Henry Gage.

3rd ed

Wissenschaftliche Ergebnisse der Deutschen Tiefsee-Expedition auf dem Dampfer "Valdivia" 1898-1899. Im Auftrage des Reichsamtes des Innern hrsg. von Carl Chun [et al.]

bd. 4, Text

Wissenschaftliche Ergebnisse der Deutschen Tiefsee-Expedition auf dem Dampfer "Valdivia" 1898-1899. Im Auftrage des Reichsamtes des Innern hrsg. von Carl Chun [et al.]

bd. 14, T. Spezieller

Wissenschaftliche Ergebnisse der Deutschen Tiefsee-Expedition auf dem Dampfer "Valdivia" 1898-1899. Im Auftrage des Reichsamtes des Innern hrsg. von Carl Chun [et al.]

bd. 14, L. 1, Tafeln

Notas de ixodologia: VI - Descrição da fêmea de Amblyomma Multipunctum, Neumann 1899 e redescrição do macho (Acari: Ixodídae)

The female of Amblyomma multipunctum Neumann, 1899 is described for the first time, the species being easily recognized by the unusual formula 5/5 in the female's hypostoma, the formula of the male's being 4/4. A redescription of the male is presented.