Hyperreal differentiation with an idempotent ultrafilter

Abstract

In the hyperreals constructed using a free ultrafilter on R , where [f] is the hyperreal represented by f:R—>R, it is tempting to define a derivative operator by [f]’=[f’], but unfortunately this is not generally well-defined.  We show that if the ultrafilter in question is idempotent and contains (0,\epsilon) for arbitrarily small real \epsilon, then the desired derivative operator is well-defined for all f such that [f’] exists. We also introduce a hyperreal variation of the derivative from finite calculus, and show that it has surprising relationships to the standard derivative. We give an alternate proof, and strengthened version, of Hindman’s Theorem.