An extension of E. Bishop's localization theorem

We prove that if f is a function belonging to Baire first class on a compac t set K subset of C and each point of K has a (closed) neighborhood where f is the pointwise limit of some sequence of uniformly bounded rational func tions, then f on the whole of K is the pointwise limit of a sequence of rat ional functions uniformly bounded on K. This is an extension of Bishop's lo calization theorem. As an application we establish a "pointwise" version of Mergelyan's classical theorem on uniform approximation by rational functio ns on compact sets for which the components of its complement have diameter s greater than a fixed positive number. (C) 2001 Academic Press.