Various Lipschitz-like Properties for Functions and Sets I: Directional Derivative and Tangential Characterizations

In this work we introduce for extended real valued functions, defined on a Banach space X, the concept of K directionally Lipschitzian behavior, where K is a bounded subset of X. For different types of sets K (e.g., zero, singleton, or compact), the K directionally Lipschitzian behavior recovers well-known concepts in variational analysis (locally Lipschitzian, directionally Lipschitzian, or compactly epi-Lipschitzian properties, respectively). Characterizations of this notion are provided in terms of the lower Dini subderivatives. We also adapt the concept for sets and establish characterizations of the mentioned behavior in terms of the Bouligand tangent cones. The special case of convex functions and sets is also studied.