Direct Proofs of the Fundamental Theorem of Calculus for the Omega Integral

TL;DR

This paper provides direct proofs of the Fundamental Theorem of Calculus for the omega integral, an extension of the Riemann integral, without relying on the Riemann integral itself.

Contribution

It offers the first direct proofs of the Fundamental Theorem of Calculus for the omega integral, enhancing understanding of this integral's properties.

Findings

Established direct proofs for continuous functions

Extended the Fundamental Theorem of Calculus to the omega integral

Clarified the relationship between omega and Riemann integrals

Abstract

When introduced in a 2018 article in the American Mathematical Monthly, the omega integral was shown to be an extension of the Riemann integral. Although results for continuous functions such as the Fundamental Theorem of Calculus follow immediately, a much more satisfying approach would be to provide direct proofs not relying on the Riemann integral. This note provides those proofs.