Chapter 3 - Lipschitz Functions and Rectifiable Sets

Abstract

This chapter introduces the m-dimensional surfaces of geometric measure theory, called rectifiable sets. These sets have folds, corners, and more general singularities. The relevant functions are not smooth functions as in differential geometry, but Lipschitz functions. A function f: Rm → Rn is Lipschitz, if there is a constant C. The least such constant is called the Lipschitz constant and is denoted by Lip f. A Lipschitz function f : Rm → Rn is differentiable almost everywhere and its proof has five steps including: a monotonic function f : Rm → Rn is differentiable almost everywhere.