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 Pawłucki's extension of the Puiseux lemma. 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap87-0-2,
author = {J\'er\^ome Bolte and Aris Daniilidis and Adrian Lewis and Masahiro Shiota},
title = {Clarke critical values of subanalytic Lipschitz continuous functions},
journal = {Annales Polonici Mathematici},
volume = {85},
year = {2005},
pages = {13-25},
zbl = {1090.35033},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap87-0-2}
}
Jérôme Bolte; Aris Daniilidis; Adrian Lewis; Masahiro Shiota. Clarke critical values of subanalytic Lipschitz continuous functions. Annales Polonici Mathematici, Tome 85 (2005) pp. 13-25. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap87-0-2/