On an integral as an interval function

TL;DR

This paper introduces a new integral definition based on total integrability and antiderivatives, linking it to the generalized Riemann integral through the concept of residues in real analysis.

Contribution

It proposes a novel integral as an interval function derived from antiderivatives and establishes its relationship with the generalized Riemann integral via residues.

Findings

Defined an integral as an interval function associated with an antiderivative.

Established the relationship between the new integral and the generalized Riemann integral.

Introduced the concept of residues into real analysis.

Abstract

Based on the total integrability we first define an integral of a real valued function f as an interval function associated to its antiderivative F. By introducing the concept of the residue of a function into the real analysis, the relationship between the so defined integral and the generalized Riemann integral is established.