The Fundamental Theorem of Calculus with Gossamer numbers

TL;DR

This paper extends the Fundamental Theorem of Calculus to the gossamer number system, which includes infinitesimals and infinities, and explores Riemann sums within this framework.

Contribution

It introduces a proof of the FTC within gossamer numbers and analyzes Riemann sums' behavior at infinity, including their non-uniqueness.

Findings

FTC holds in the gossamer number system.

Riemann sums can be represented as continuous functions with infinitesimal intervals.

Riemann sums exhibit non-uniqueness at infinity.

Abstract

Within the gossamer numbers which extend the real numbers to include infinitesimals and infinities we prove the Fundamental Theorem of Calculus (FTC). Riemann sums are also considered in the gossamer number system, and their non-uniqueness at infinity. We can represent the sum as a continuous function in the gossamer number system by inserting infinitesimal intervals at the discontinuities, and threading curves between the sums discontinuities. As the FTC is a difference of integrals at the end points, the same is true of sums.