10. Generalized gradients

Let \(f:X\to\,{\mathbb{R}}\) be a lower semicontinuous function defined on a Banach space X. If f is continuously differentiable, then the mean value theorem implies that it is locally Lipschitz. If, instead, the function f is convex, then once more it has the property of being locally Lipschitz, as we have seen. In this sense, the class of locally Lipschitz functions subsumes the smooth and convex cases. The class of locally Lipschitz functions has other features that recommend it as an environment in which to develop a theory of nonsmooth calculus, which is our goal in this chapter. It is closed under familiar operations such as sum, product, and composition. But it is also closed under less classical ones, such as taking lower or upper envelopes. Finally, it includes certain nonsmooth, nonconvex functions that are important in a variety of applications, notably distance functions. We proceed to develop the calculus of the generalized gradient of a locally Lipschitz function f, denoted ∂ _C f(x). This leads to a unified treatment of smooth and convex calculus, as well as an associated geometric theory of tangents and normals to arbitrary closed sets.