Publisher's Synopsis
In the study of the derivation properties of interval functions, there are certain arguments that reappear in many settings. In this book, the author seeks to present a unified approach to some of these techniques. The motivation grows out of the interesting and important study of Rogers and Taylor characterizing those interval functions which are, in a sense, absolutely continuous with respect to the 8-dimensional Hausdorff measure. This problem leads naturally to an investigation of Lipschitz numbers Ds(f,x) = lim sup y,z?x,yz(f(z)-f(y))/(z-y)s and to 8-dimensional integrals. The exposition is presented in the setting of interval functions on the real line and the differentiation, measure-theoretic, and variational properties are developed. The author limits attention to the one-dimensional case; many of the arguments can be used in higher dimensions, but the richer geometry and larger choice of differentiation bases are apt to obscure the natural simplicity of the ideas. Also presented are applications to the Hausdorff and packing measures, as well as to the classical differentiation theory of real functions.
