Hyperreal differentiation with an idempotent ultrafilter

TL;DR

This paper explores the conditions under which a derivative operator can be well-defined in hyperreal numbers constructed via ultrafilters, introducing a new hyperreal derivative and relating it to standard derivatives.

Contribution

It demonstrates that with an idempotent ultrafilter containing small neighborhoods, the hyperreal derivative becomes well-defined, and introduces a hyperreal derivative variant linked to finite calculus.

Findings

Derivative operator is well-defined under specific ultrafilter conditions

Introduces a hyperreal variation of the derivative from finite calculus

Provides an alternative proof and strengthening of Hindman's theorem

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.