This article is concerned with the continuous triangularization of matrix functions which depend continuously on several variables. By use of an algorithm analogous to the one employed for the reduction of a $\lambda $-matrix to a diagonal form, we find a continuous similarity transformation which produces the triangularization of a given matrix.Let there be given a singular system of differential equations whose coefficient matrix depends solely on several small parameters. Then, our method may be applied to obtain the complete asymptotic expansion of a fundamental matrix solution of the singular differential system at its singular point.