Critical points of simple functions

Abstract

Among various notions of critical points available for nonsmooth functions, the approach via the ‘weak slope’ has considerable appeal. This property is less restrictive than that based on the ‘strong’ slope of DeGiorgi–Marino–Tosques, but remains purely metric in nature. Within variational analysis, this class of critical points is intermediate between those associated with the Clarke and the limiting subdifferentials. However, recognizing such points for concrete functions seems challenging. We present a basic topological characterization for the simplest nontrivial case: piecewise affine (and, more generally, ‘definable’) functions of two variables.

Keywords:

• nonsmooth critical point
• weak slope
• subdifferential

AMS 2000 Subject Classifications: :

• 49J52
• 58E05