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arXiv · 2411.14689

Hyperreal differentiation with an idempotent ultrafilter

Samuel Allen Alexander·Bryan Dawson
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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.

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Samuel Allen Alexander, Bryan Dawson. 2024-11-22. Hyperreal differentiation with an idempotent ultrafilter. https://doi.org/10.4115/jla.2024.16.6

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