Hyperreal differentiation with an idempotent ultrafilter

Abstract

For every f:R->R, the sequence obtained by restricting f to the positive integers is a representative of a hyperreal number [f]. If f is differentiable, it is natural to consider the hyperreal [f'] represented by f'. Unfortunately, there exist differentiable functions f,g:R->R such that [f] equals [g] but [f'] is not equal to [g'], so the operation of taking derivatives does not induce the function on hyperreals that one might want it to. Using idempotent ultrafilters, we introduce a new derivative notion, assigning a hyperreal-valued function Df:R->*R to every f:R->R (even when f is not differentiable), such that [f] equals [g] implies [Df] equals [Dg] and thus inducing a mapping D([f]) equals [Df]. In a technical sense, our Df is infinitely close to f' provided f is differentiable and 2pi-periodic.