Integration of Delta Continuous Stieltjes Variational in the Plane

TL;DR

This paper explores the properties of the delta continuous Stieltjes variational integral in the plane, generalizing the integral concept from the real line and examining its primitive functions as interval functions.

Contribution

It introduces fundamental properties of the delta continuous Stieltjes variational integral in the plane, extending the concept from the real line and analyzing its primitive functions.

Findings

Generalization of the integral to the plane

Primitive functions as interval functions

Fundamental properties established

Abstract

This paper deals with the delta continuous Stieltjes variational integral generalized in the plane. In particular, this work presents about some fundamental properties of it. The delta continuous Stieltjes variational integral in the plane is considered as a form of generalization of the same type of integral defined in the real line [4]. The extremely difference between the delta continuous Stieltjes variational integrals defined in both the real line and the plane is especially related to their primitive functions. Where, if $F$ is a primitive function of the delta continuous Stieltjes variational integrable $f$ defined on the real line, then $F$ is a point function. Meanwhile, in the plane the function $F$ is looked at as an interval function. Therefore, in order to investigate the properties of the integration of delta continuous Stieltjes variational in the plane is often needed some ideas appeared previously (see 1. Introduction) which are not available in [4].