Critical values of set-valued maps with stratifiable graphs. Extensions of Sard and Smale-Sard theorems

We prove three theorems extending Sard’s theorem and its infinite dimensional extension due to Smale to set-valued mappings with stratifiable graphs. The very concept of a critical value comes from (nonsmooth) variational analysis and turns out to be perfectly compatible with the natural condition defining “good” stratifications (e.g., Whitney stratification in the finite dimensional case).