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

Abstract In this paper we explore the concept of locally band preserving functions, introduced by Ercan and Wickstead, on Dedekind complete $$\Phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Φ</mml:mi> </mml:math> -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 $$\Phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Φ</mml:mi> </mml:math> -algebras, a complex version of the Mean Value Theorem is also provided.