This manuscript discusses the structure of the errors in the solution of elliptic problems introduced by the regularization of the fluid properties discontinuity in a small region of finite size. By using a multiscale approach, the problem of the error calculation is splitted into an outer problem, where the regularization region is replaced by a sharp interface, and an inner local one dimensional problem that eventually imposes effective jump conditions across the interface for the outer problem. Except in some particular cases, the use of regularization techniques introduces first order errors in the solution imposing an error flux jump that is proportional to the tangential Laplacian of the averaged solution and an error jump that is proportional to the normal flux, both multiplied by a prefactor that depends on the averaging rule used. In general, the optimal averaging procedure is shown to depend on the structure of the problem at hand. The errors introduced by standard arithmetic and harmonic averages are obtained for various analytical and numerical examples which are used to discuss the nature of the errors introduced and the importance of first order errors in the solution. The influence of the ratio between the regularization thickness and the grid size is also investigated in numerical implementations of the one-fluid model.