Classical theorems from analysis for locally band preserving functions on Dedekind complete $$\Phi $$-algebras

Positivity, vol. 30

Abstract

Abstract In this paper we explore the concept of locally band preserving functions, introduced by Ercan and Wickstead, on Dedekind complete $$\Phi $$ Φ -algebras. Specifically, we show that all super order differentiable functions are locally band preserving. Furthermore, some foundational results from classical analysis are proved in this setting, such as the Intermediate Value Theorem, the Extreme Value Theorem, and the Mean Value Theorem. Moreover, we show that these generalisations can fail for functions that are not locally band preserving. With the goal in mind to further develop the theory of complex differentiation in Dedekind complete complex $$\Phi $$ Φ -algebras, a complex version of the Mean Value Theorem is also provided.

Authors 4

1. Eder Kikianty

   University of the Witwatersrand

   Affiliation as printed

   School of Mathematics, University of the Witwatersrand, Private Bag 3, 2050, WITS, South Africa

2. Luan Naude

   University of Pretoria

   Affiliation as printed

   Department of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X20, 0028, Hatfield, South Africa

3. Mark Roelands Aachen Mathematical Institute

   Leiden University

   Affiliation as printed

   Mathematical Institute, Leiden University, 2300 RA, Leiden, The Netherlands

4. Christopher Schwanke corresponding

   University of Pretoria

   Affiliation as printed

   Department of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X20, 0028, Hatfield, South Africa