In earlier papers, classes of transfer policies have been studied and maximal and minimal Lorenz curves   obtained. In addition, there are policies belonging to the class with given Gini indices or passing through given points in the  plane. In general, a transformation  describing a realistic transfer policy has to be continuous. In this paper the results are generalized and the class of transfer policies  is modified so that the members may be discontinuous. If there is an optimal policy which Lorenz dominates all policies in the class, it must be continuous. The necessary and sufficient conditions under which a given differentiable Lorenz curve  can be generated by a member of a given class of transfer policies are obtained. These conditions are equivalent to the condition that the transformed variable  stochastically dominates the initial variable X. The theory presented is obviously applicable in connection with other income redistributive studies such that the discontinuity can be assumed. If the problem is reductions in taxation, then the reduction for a taxpayer can be considered as a new benefit. The class of transfer policies can also be used for comparisons between different transfer-raising situations.