There are very weak definitions of continuity available for series. Of course, one can easily construct functions that are continuous but not differentiable. However, the majority of USEFUL constructs if they are defined continuously ARE differentiable.

In this case, why prefer to use only difference equations, as Y.C. said above in the comments? They are not easier to work with. The easiest equations to work with are complex differentiable, as Painleve once said. Furthermore, it is questionable whether the real world is discrete. We don't know. The equations in quantum mechanics are continuous or differences of continuous equations.

Also, modern topological foundations operate with continuous concepts primarily. If we do no require continuity, we cannot understand many restrictions on what is and is not possible.

A great illustration is the book : C. Truesdell, 1948, *A Unified Theory of Special Functions, based upon the functional equation $\frac{\partial}{\partial z}F(z,a)=F(z,a+1)$*, the Princeton University Press

It managed to eliminate much useless specialization by revealing a general complex continuous fact that could be used to construct a general method of expressing almost any special function in terms of another one arbitrarily chosen.

EDIT:

K.C. in comments above also says much of what I was going to add here. He should post it as an answer? For this is an important reason why $\mathbb{R}$ is used in physical models.

The substance of the argument can be found in Leibnitz's 1676 papers (which also introduced the diagonal argument and 1-to-1 correspondence as measures of length!)

Seeing how much of the discussion seems to admit that mathematics models nature only approximately view (NOA-V), I would like to give a (representative?) reference for the reasons supporting the opposite viewpoint (call it the Leibnitz-Euler-Dirac View = LED-V):

C. Truesdell, 1966, Method and Taste in Natural Philosophy, *Six Lectures in Natural Philosophy*, Springer.

This book *Six Lectures* is misleading in it's title and is almost pure mathematics, not philosophy. It an even better example of the power of continuity in models, a brilliant use of continuity to solve ergodicity questions for phase spaces with finite states.

definitionis simpler this does not make them easier towork with$\endgroup$ – Yemon Choi Sep 29 '14 at 18:33Furthermore, difference equations don't require complicated epsilon-delta definitions.Nor do differential equations require epsilon-delta definitions. Newton and Leibniz solved differential equations about 150 years before epsilontics came along. $\endgroup$ – Ben Crowell Sep 30 '14 at 3:46