The Fundamental Theorem of Calculus with Gossamer numbers

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.