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In the theory of the boundary behaviour of canonical conforma,l mappings on open Riemann surfaces, it is a well known fact that every canonical conformal mapping on an open Riemann surface of finite genus is a vertical slit mapping ([4], [6], [9]). This theorem was first proved by M. Mori ([7]). For the proof of the theorem, she prepared a lemma (Lemma 2, p. 177) which asserts that every meromorphic function / of bounded valence on an open Riemann surface .B of finite genus has a limit at each weak boundary point of the Kerdkjårtd-Stoilow boundary of .8. However, P. Järvi ([3]) showed that the assertion of the lemma contains invalid arguments. Although he could not restore the assertion of Mori, he proved that the lemma holds true if R is an open Riemann surface with absolutely disconnected boundary (see [8]) and if no cluster set of / at Ker6kjårt6-Stoilow boundary point of R separates the extended complex plane. He further pointed out an error in the example introduced by Kusunoki and Taniguchi ([5]) who intended to show that frf=rO,qo,n Ono I a. Moreover, he remarked that one can obtain a planar Riemann surface of the class flp1oAD,n Oap provided that Mori's assertion holds true for planar Riemann surfaces. In this note, we shall first give a simple example of a Riemann surface to show that Mori's assertion does not hold even for planar Riemann surfaces. Secondly, we shall prove that every meromorphic function of bounded valence on an open Riemann surface .E wiih absolutely disconnected boundary admits a continuous extension to the Kerdkjårtd-Stoilow boundary of .R. Consequently, according to the proof of Järvi ([3]), we can conclude that there exists a planar Riemann surface which belongs to the class flpro AD,n O,s.o.

Meromorphic functions of bounded valence on an open Riemann surface