> Completeness gives you limits, the simplest workable definitions of the derivative and integral, and the differentiability of many important classes of functions.
Nit-picking can be continued endlessly, but, as a final salvo, the definitions of limits, derivatives, and integrals don't depend on completeness (which is a good thing in the first case, since the (uniform-space, as opposed to order-theoretic) notion of completeness depends on that of limits). As you say, the existence of certain limits and integrals needs completeness. (I don't know off the top of my head any derivatives that one needs completeness to compute—rather nice consequences of derivatives, like that only constant functions have 0 derivative—but that's probably my ignorance, rather than a genuine lack.)
Nit-picking can be continued endlessly, but, as a final salvo, the definitions of limits, derivatives, and integrals don't depend on completeness (which is a good thing in the first case, since the (uniform-space, as opposed to order-theoretic) notion of completeness depends on that of limits). As you say, the existence of certain limits and integrals needs completeness. (I don't know off the top of my head any derivatives that one needs completeness to compute—rather nice consequences of derivatives, like that only constant functions have 0 derivative—but that's probably my ignorance, rather than a genuine lack.)