<![CDATA[The Mean Value Theorem (MVT) is a fundamental result in real analysis, connecting the average rate of change of a function with its instantaneous rate of change. Classical formulations require continuity on a closed interval and differentiability on the corresponding open interval. This paper investigates the minimal assumptions underlying the MVT, establishes equivalent formulations, and analyzes their logical relationship with differentiability on closed intervals. The study employs theoretical analysis, logical deduction, and counterexample construction. The results demonstrate that differentiability on an open interval automatically guarantees continuity at every interior point. Consequently, continuity assumptions can be rigorously reduced to endpoint continuity alone. Furthermore, this study provides an analytical proof that the Minimal Formulation of the MVT is not equivalent to differentiability on the closed interval [a,b] due to the overly restrictive nature of endpoint derivatives. Three nested formulations of the MVT are established, alongside an analytical characterization of topological conditions leading to its failure]]>
Copyrights © 2026
