The paper by Frank H. Clarke develops a theory of generalized gradients for a broad class of functions and corresponding normals to arbitrary closed sets. These concepts subsume the usual gradients and normals of smooth functions and manifolds, as well as the subdifferentials and normals of convex analysis. A key result is a theorem concerning the differentiability properties of functions of the form \(\max \{ g(x, u) : u \in U \}\), which unifies and extends previous theorems by Danskin and others. The theory is then applied to characterize flow-invariant sets, yielding corollaries of the theorems by Bony and Brezis. The introduction highlights the importance of replacing smoothness assumptions with convexity in optimization, and the paper provides detailed proofs and examples to illustrate the theory.The paper by Frank H. Clarke develops a theory of generalized gradients for a broad class of functions and corresponding normals to arbitrary closed sets. These concepts subsume the usual gradients and normals of smooth functions and manifolds, as well as the subdifferentials and normals of convex analysis. A key result is a theorem concerning the differentiability properties of functions of the form \(\max \{ g(x, u) : u \in U \}\), which unifies and extends previous theorems by Danskin and others. The theory is then applied to characterize flow-invariant sets, yielding corollaries of the theorems by Bony and Brezis. The introduction highlights the importance of replacing smoothness assumptions with convexity in optimization, and the paper provides detailed proofs and examples to illustrate the theory.