Clarke critical values of subanalytic Lipschitz continuous functions

Volume 87 / 2005

Jérôme Bolte, Aris Daniilidis, Adrian Lewis, Masahiro Shiota Annales Polonici Mathematici 87 (2005), 13-25 MSC: Primary 35B38; Secondary 49J52, 32B30. DOI: 10.4064/ap87-0-2


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.


  • Jérôme BolteÉquipe Combinatoire et Optimisation (UMR 7090, Case 189)
    Université Pierre et Marie Curie
    4 Place Jussieu, 75252 Paris Cedex 05, France
  • Aris DaniilidisDepartament de Matemàtiques, C1/320
    Universitat Autònoma de Barcelona
    E-08193 Bellaterra (Cerdanyola del Vallès), Spain
  • Adrian LewisSchool of Operations Research and Industrial Engineering
    Cornell University
    234 Rhodes Hall
    Ithaca, NY 14853, U.S.A.
  • Masahiro ShiotaDepartment of Mathematics
    Nagoya University (Furocho, Chikusa)
    Nagoya 464-8602, Japan

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