The Fundamental Theorem of Calculus: A Duality in Non-Smooth Analysis

This paper explores the extension of the Fundamental Theorem of Calculus (FTC) to the domain of non-smooth analysis. The classical FTC, which establishes a profound duality between differentiation and integration for smooth functions, breaks down when differentiability is not guaranteed. Non-smooth analysis provides a rigorous framework to generalize concepts of derivatives to non-differentiable functions, primarily through the use of subdifferentials and generalized gradients. This work investigates how the FTC can be reformulated in this broader context, revealing a persistent, albeit more complex, duality. We examine the theoretical underpinnings required for this extension, including the properties of locally Lipschitz functions, the calculus of set-valued mappings, and the appropriate notions of integration, such as the Aumann integral. The core of the paper demonstrates that for a non-smooth function, the integral of its subdifferential over an interval contains the total change in the function's value, expressed as an inclusion rather than an equality. This set-valued generalization restores the essential link between differentiation and integration. The implications of this non-smooth duality are discussed, particularly in application areas such as optimization theory, optimal control, and mechanics, where non-differentiable functions are frequently encountered.