A notion of primitive that fits the Riemann integral

TL;DR

This paper introduces a new concept of primitive functions that precisely match the Riemann integral, leading to a clear characterization of Riemann integrability and a simplified presentation of the Fundamental Theorem of Calculus.

Contribution

It defines a primitive notion aligned with the Riemann integral, providing a direct characterization of integrability and a streamlined approach to the Fundamental Theorem of Calculus.

Findings

Characterization of Riemann integrability through the new primitive concept

A generalized lemma on functions with zero derivative

A simplified presentation of the Riemann integral in R

Abstract

We present a notion of primitive which corresponds exactly with the Riemann integral. We obtain a characterization of the integrability in the sense of Riemann which produces a Fundamental Theorem of Calculus without special assumptions. We prove a lemma which generalizes the theorem on functions with zero derivative in an interval. We apply these concepts, sketching a simplified alternative presentation of the Riemann integral in R.