# The Sard theorem for essentially smooth locally Lipschitz maps and applications in optimization

The classical Sard theorem states that the set of critical values of a \$C^{k}\$-map from an open set of \$\R^n\$ to \$\R^p\$ (\$n\geq p\$) has Lebesgue measure zero provided \$k\geq n-p+1\$. In the recent paper by Barbet, Dambrine, Daniilidis and Rifford, the so called ``preparatory Sard theorem" for a compact countable set \$I\$ of \$C^k\$ maps from \$\R^n\$ to \$\R^p\$ and a Sard theorem for a locally Lipschitz continuous selection of this family have been established under the assumption that \$k\geq n-p+1\$. Here, we show that, in the special case \$n=p\$ and \$I\$ is finite, the \$C^1\$ smoothness assumption in these results can be relaxed to ``essentially smooth locally Lipschitz". Then we apply the obtained results to study Karush-Kuhn-Tucker type necessary condition for scalar/vector parametrized constrained optimization problems and the set of Pareto optimal values of a continuous selection of a finite family of essentially smooth locally Lipschitz maps.