Hilbert [13] suggested in connection with his fifth problem that, while the theory of differential equations provides elegant and powerful techniques for solving functional equations, the inherent differentiability assumptions are typically unnatural (see [4]). Such shortcomings can sometimes be overcome by appealing to regularity theorems of the type "continuity implies differentiability" (see e.g. [4], and w of [1]) and of the type "measurability implies continuity" beginning with Fr~chet [9] and continued by scorses of others (see e.g. [4], [5], [16] and [19] for references to further such results). The theory of distribution of Laurent Schwartz has also been used, perhaps first by Feny5 [8], (see also [2], [6], [11], [12], [20] and [25]) to partially overcome this deficiency. The main aim of the present paper is to introduce another distributional method, based on approximating distributional derivatives by differences, which allows one to find the distributional solutions to distributional analogues of many functional equations by reduction to (distributional) differential equations. This method, when combined with regularity theorems, allows one to find all the Continuous (or even locally Lebesgue integrable) solutions of certain functional equations. References to some of the vast literature concerning many of the functional equations considered herein can be found in [1] and [3]. While many of our results concerning functional equations may be considered as bu"t slight generalizations of known results, our main aim is to illustrate a method of solution. A novelty of our approach is that it permits "local" consideration. Unless otherwise indicated, our notation and terminology is that of Rudin [24]. The set of all Schwartz test function on R n will be denoted by 79~. We let D~ denote the set of Schwartz distributions on R n. All measure-theoretic language refers to Lebesgue measure.