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<p>In this paper we explore the concept of locally band preserving functions, introduced by Ercan and Wickstead, on Dedekind complete <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\Phi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation>-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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\Phi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation>-algebras, a complex version of the Mean Value Theorem is also provided.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月14日

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  • Eder Kikianty,
  • Luan Naude,
  • Mark Roelands,
  • Christopher Schwanke

摘要

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.