ABSTRACT
The notion of a covering theorem and its role in the study of the differentiation properties of functions are now widely known. In many parts of modern analysis these concepts are now central. The same program can be applied equally well to the study of functions of a single variable with a resulting clarification of techniques. This chapter develops the basic covering theorems that should be associated with any study of the various symmetry properties of real functions. It clarifies a considerable economy afforded by this approach and many proofs. The chapter expresses the Vitali Theorem in an imprecise form in order to illustrate it as belonging to the same family of ideas as the more elementary theorem which follows.
