The argument principle and holomorphic extendibility to finite Riemann surfaces

Let M be a finite Riemann surface and let A(M) be the algebra of all continuous functions on M∪bM which are holomorphic on M. We prove that a continuous function Φ on bM extends to a function in A(M) if and only if for every f,g in A(M) such that fΦ+g≠0 on bM, the change of argument of fΦ+g is nonnegative.