ABSTRACT
This chapter turns to a problem that arises from the same type of concern. It asks for the continuity properties of a function that satisfies some kind of symmetric growth condition. The chapter introduces the reader to the subject nearly a complete picture of what is known; and offers some refinements that require more technical apparatus than prepared to develop for the moment. It devotes to related concerns and addresses a problem that remains open: to determine the exact nature of the set of points of discontinuity of a symmetrically continuous function. The theorem of Stein and Zygmund that state and prove shows that, under a measurability assumption, symmetric continuity and ordinary continuity are equivalent up to a set of measure zero.
