Introduction. The abstract varieties (also called manifolds) of difference algebra [2], [3] have not heretofore had a realization as sets of functions comparable to the realization provided for differential manifolds by the analytic existence theorem for differential equations (see [4], particularly p. 23). It is the purpose of this Note to provide such a realization by means of an existence theorem yielding solutions of difference equations as complex-valued functions defined, except for isolated singularities, on the non-negative real axis. Since the abstract varieties consist by definition of elements lying in integral domains, these functions are required to generate difference rings which are integral domains. This distinguishes the existence theorem from the mere use of the difference equation as a recursion relation. In addition, the functions studied here are piecewise analytic, though discontinuous. There is no reason to regard the class of functions selected in this Note as definitive: other choices may better repay further study. It would, in particular, be interesting to know whether one can obtain continuous solutions. This question is discussed briefly in the next to last section. The concluding section provides a partial analogue of an important approximation theorem of differential algebra ([4, p. 123]). The notation and terminology are as in [3], and an earlier discussion of the problem with particular reference to meromorphic solutions will be found there, on pp. 114 and 242. It might seem natural to seek solutions defined only on the integers, rather than on a half-line. But this is not possible. The only solutions of the difference polynomial yyl-1 which are complex-valued functions on the integers and generate difference rings which are integral domains are also solutions of y2 -1. Yet the abstract variety of yyl -1 is not contained in that of y2 1. Alternatively, it is very likely possible to obtain suitable solutions as functions from the integers to a ring of functions of one or more variables. This amounts to finding solutions defined on the integers and involving arbitrary constants.