(1.1) |f(a)− f(b)| ≤ L |a− b| for every pair of points a, b ∈ A. We also say that a function is Lipschitz if it is L-Lipschitz for some L. The Lipschitz condition as given in (1.1) is a purely metric condition; it makes sense for functions from one metric space to another. In these lectures, we concentrate on the theory of Lipschitz functions in Euclidean spaces. In Section 2, we study extensio...

We show that in two dimensions or higher, the Mordukhovich-loffe approximate subdifferential and Clarke subdifferential may differ almost everywhere for real-valued Lipschitz functions. Uncountably many Frechet differentiable vector-valued Lipschitz functions differing by more than constants can share the same Mordukhovich-Ioffe coderivatives. Moreover, the approximate Jacobian associated with ...

We study rank functions (also known as graph homomorphisms onto Z), ways of imposing graded poset structures on graphs. We first look at a variation on rank functions called discrete Lipschitz functions. We relate the number of Lipschitz functions of a graph G to the number of rank functions of both G and G× E . We then find generating functions that enable us to compute the number of rank or L...

Let M be an o-minimal structure over the real closed field R. We prove the definable smoothing of definable Lipschitz continuous functions. In the case of Lipschitz functions of one variable we are even able to preserve the Lipschitz constant.