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In this manuscript, we provide a generalization of the Cauchy integral to appropriate families of real-valued functions defined in. The results are motivated by a generalization of regulated functions. The foundational theory for this extended Cauchy integral is developed in detail in this self-contained work. We derive some properties for appropriate families of functions and introduce the Cauchy integral for appropriate step functions. Using these results, the integral for generalized regulated functions is presented next together with some basic properties of the generalized Cauchy integral, including the linearity, the non-negativity and the monotonicity properties. Various illustrative examples on the applicability of our theoretical results are provided.

A foundational theory of an extended Cauchy integral for generalized regulated real-valued functions on