Real Functions and its Differentiation in Alternative Set Theory

TL;DR

This paper extends previous work on real numbers in alternative set theory by exploring real functions and their differentiation, showing that fundamental properties are preserved in AST similar to classical calculus.

Contribution

It introduces a framework for real functions and differentiation within AST, maintaining core properties akin to standard calculus, thus broadening the scope of AST.

Findings

Differentiation properties in AST mirror classical calculus

Real functions in AST exhibit expected continuity and differentiability behaviors

The framework supports further analysis of real functions in alternative set theory

Abstract

In the previous paper (Kiri Sakahara and Takashi Sato. Basic Topological Concepts and a Construction of Real Numbers in Alternative Set Theory. arXiv e-prints, arXiv:2005.04388, May 2020), the authors displayed basic topological concepts and a construction of a system of real number in alternative set theory (AST). The present paper is a continuation of that research providing additional treatments of real functions. The basic properties of differentiation in AST are preserved as in the conventional calculus.