Nonsmooth Morse–Sard theorems

Abstract

We prove that every function f:Rn→R satisfies that the image of the set of critical points at which the function f has Taylor expansions of order n−1 and non-empty subdifferentials of order n is a Lebesgue-null set. As a by-product of our proof, for the proximal subdifferential ∂P, we see that for every lower semicontinuous function f:R2→R the set f({x∈R2:0∈∂Pf(x)}) is L1-null.