Work

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

Year: 2026

Type: article

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.

Source: Positivity

DOI: 10.1007/s11117-026-01188-6

Authors: Eder Kikianty, Luan Naude, Mark Roelands, Christopher Schwanke

Institutions: University of the Witwatersrand, University of Pretoria, Leiden University

Language: English

Topic: Advanced Banach Space Theory

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,090

Funders: University of Pretoria, Universiteit Leiden