A Corollary on the Local Differentiability of Riemann Integrals via Barrow's Rule

We present a pedagogical note on the First Fundamental Theorem of Calculus (1FTC), exploring its extension to a class of discontinuous functions using the hypotheses of Barrow's Rule (2FTC). Operating strictly within Riemann integration theory, we utilize a local incremental quotient framework to deduce the classical derivative-integral relationship. Furthermore, we address the mathematical folklore surrounding the logical circularity of this inversion. By applying Darboux's Theorem and analyzing the classic bounded oscillatory discontinuity, we demonstrate that continuity is a sufficient but not a necessary condition for the local validity of the fundamental relation, offering an insightful perspective for undergraduate analysis courses.