Functions of Several Variables

Abstract

Because of the central role played by absolutely continuous functions and functions of bounded variation in the development of the fundamental theorem of calculus in $\mathbb {R}$, it is natural to ask whether they have analogues among functions of more than one variable. One of the main objectives of this chapter is to show that this is true. We have found that the BV functions in $\mathbb {R}$ constitute a large class of functions that are differentiable almost everywhere. Although there are functions on $\mathbb {R}^{n}$ that are analogous to BV functions, they are not differentiable almost everywhere. Among the functions that are often encountered in applications, Lipschitz functions on $\mathbb {R}^{n}$ form the largest class that are differentiable almost everywhere. This result is due to Rademacher and is one of the fundamental results in the analysis of functions of several variables.