On the continuation of meromorphic functions

PENTTT.lÄnvr l.Let W be an open Riemann surface and let V be an arbitrary end of W, i.e., a subregion of If wil"h compact relative boundary \y,V.Let MC(V) denote the class of meromorphic functions on Z which have a limit at every point of the relative Stoilow ideal boundary Py of Z. Further, let BY(V) denote the class of constants and of meromorphic functions of bounded valence on Y, atdlet MD* (V) stand for the class of meromorphic functions with a finite spherical Dirichlet integral on Z.As in [3] and [4], we are interested in the interrelations of these three classes on certain Riemann surfaces.In Section 2, we discuss the possibility of obtaining BY(V)cMC(V) ot Riemann surfac,es whose boundaries are absolutely disconnect- ed in the sense of Sario [13].In Section 3, we seek conditions upon B under which