Mean Value Theorems and L'Hospital-Type Rules for Regulated Functions

TL;DR

The paper extends classical L'Hospital-type rules to regulated functions by introducing the $D_{\alpha}$ derivative, defined relative to a strictly increasing $\alpha$, unifying continuous and discrete (Stolz-Cesaro) settings and linking to Lebesgue-Stieltjes integrals. It establishes a regulated-function Cauchy mean value theorem and derives generalized L'Hospital and monotone rules with endpoint behavior, including cases where one-sided derivatives or discontinuities arise. A measure-theoretic version via Lebesgue-Stieltjes integrals is developed and compared with the $D_{\alpha}$ framework, together with illustrative examples. Overall, the work provides a unified, robust toolkit for mean-value and quotient-rule analysis beyond classical smoothness assumptions, with broad applicability to problems featuring non-smooth or jump discontinuities.

Abstract

We introduce a generalization of Cauchy's mean value theorem for regulated functions. Building on this, we extend both L'Hospital's rule and L'Hospital's monotone rule to quotients of regulated functions. We demonstrate that our extended L'Hospital's rule encompasses both the discrete case, known as the Stolz-Cesaro theorem, and the classical continuous case. In addition, we show that these extensions handle some problems that classical rules cannot address. Finally, we provide Lebesgue-Stieltjes versions of L'Hospital's rule and L'Hospital's monotone rule and compare them with our extensions.

Theorems & Definitions (27)

• Theorem 2.1
• Theorem 3.2
• proof
• Remark 3.3
• Theorem 3.4: Rolle's Theorem for Regulated Functions
• proof
• Theorem 3.5: Cauchy’s Mean Value Theorem for Regulated Functions
• proof
• Corollary 3.6
• proof