One general approach to unifying differential and difference equations goes by the term time scale calculus. From Wikipedia: "Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice — once for differential equations and once again for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale (also known as a time-set), which may be an arbitrary closed subset of the reals. In this way, results apply not only to the set of real numbers or set of integers but to more general time scales such as a Cantor set."
Besides the Wikipedia article, a standard reference is Dynamic Equations on Time Scales, by Martin Bohner and Allan Peterson. The field is relatively new - Wikipedia says it was introduced in 1988 by Stefan Hilger in (I believe, although Wikipedia does not say this) his doctoral dissertation - and so it is not that well-known yet.