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

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.