In this study we reconsider the effect of variable transformations on the redistribution of income. Under the assumption that the theorems should hold for all income distributions, earlier given conditions are both necessary and sufficient. Different versions of the conditions are compared. We also consider the consequences if we drop the explicit continuity restriction on the transformations. One main result is that continuity is a necessary condition if one pursues that the income inequality should remain or be reduced. In our earlier studies concerning tax policies the assumption that differentiable transformations satisfying a derivative condition could be reduced to transformations satisfying only a continuity condition.