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

TL;DR

This paper extends classical analysis theorems to Dedekind complete $\

Contribution

It demonstrates that super order differentiable functions are locally band preserving and generalizes key theorems like the Mean Value Theorem within this algebraic setting.

Findings

Super order differentiable functions are locally band preserving.

Classical theorems such as the Intermediate Value, Extreme Value, and Mean Value Theorems are valid in Dedekind complete $\

A complex version of the Mean Value Theorem is established.

Abstract

In this paper we explore the concept of locally band preserving functions, introduced by Ercan and Wickstead, on Dedekind complete $\Phi$-algebras. Specifically, we show that all super order differentiable functions are locally band preserving. Furthermore, some foun- dational 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 pre- serving. With the goal in mind to further develop the theory of complex differentiation in Dedekind complete complex $\Phi$-algebras, a complex version of the Mean Value Theorem is also provided.