LIPSCHITZ CONTINUATIONS OF LINEARLY BOUNDED-FUNCTIONS

The problem of the continuation of a real-valued function from a subse t Y of a metric space (X, d) to the whole of the space is considered. A well-known result of McShane enables one to extend a uniformly conti nuous function preserving its modulus of continuity. However, some nat ural questions remain unanswered in the process. A new scheme for the extension of a broad class of functions, including bounded and Lipschi tz functions, is proposed. Several properties of these extensions, use ful in applications, are proved. They include the preservation of cons traints on the increments of a function defined in terms of quasiconca ve majorants. This result enables one to refine and generalize well-kn own results on the problem of the traces of functions with bounded gra dient. The extension in question is used in two problems on function a pproximation. In particular, a direct proof of the density of the clas s Lip(X) in lip(X, omega) is given.