In Refs. [1], [2] and [4], we discussed the discontinuous Riemann-hilbert boundary value problems for nonlinear uniformly elliptic complex equations of first order in a multiply connected plane domain, but, there the solution for the modified boundary value problem with the indeces K<-1 has a pole of degree in the domain. In this paper, we give a new well posedness of the discontinuous Riemann-hilbert problem for elliptic complex equation of first order, so that its solution will be continuous in the domain. One of the significances of the paper is that the present proof of solvability of the modified boundary value problem for analytic functions and generalized analytic functions is new and simpler, which is mainly based on the zero property of analytic functions and a priori estimates of solutions and some other arguments without using the method of integral equations.