Generalizations of Rolle’s Theorem

Alberto Fiorenza () and Renato Fiorenza
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Alberto Fiorenza: Dipartimento di Architettura, Università di Napoli, Via Monteoliveto 3, 80134 Napoli, Italy
Renato Fiorenza: Accademia di Scienze Fisiche e Matematiche, Via Mezzocannone, 8, 80134 Napoli, Italy

Mathematics, 2024, vol. 12, issue 14, 1-12

Abstract: The classical Rolle’s theorem establishes the existence of (at least) one zero of the derivative of a continuous one-variable function on a compact interval in the real line, which attains the same value at the extremes, and it is differentiable in the interior of the interval. In this paper, we generalize the statement in four ways. First, we provide a version for functions whose domain is in a locally convex topological Hausdorff vector space, which can possibly be infinite-dimensional. Then, we deal with the functions defined in a real interval: we consider the case of unbounded intervals, the case of functions endowed with a weak derivative, and, finally, we consider the case of distributions over an open interval in the real line.

Keywords: Rolle’s theorem; Lagrange’s theorem; classic derivative; weak derivative; locally convex topological Hausdorff vector space; Gateaux differential; distributions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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