A mean value theorem for metric spaces

Abstract

We present a form of the Mean Value Theorem (MVT) for a continuous function f between metric spaces, connecting it with the possibility to choose the ɛ↦δ(ɛ) relation of f in a homeomorphic way. We also compare our formulation of the MVT with the classic one when the metric spaces are open subsets of Banach spaces. As a consequence, we derive a version of the Mean Value Propriety for measure spaces that also possesses a compatible metric structure.

A Polylogarithm Solution to the Epsilon--Delta Problem

“…Let us begin this section briefly recalling some preliminary tools and ideas that were addressed and proved in [4].…”

“…From the above considerations, we point a couple of interesting results that better describe all the possible topological configurations for the sets discussed above. The proofs of these theorems can be found in [4].…”

“…Theorem 19 (cf. [4]). Suppose that f : M 1 → M 2 is any given function that satisfies the Lagrange Propriety at x ∈ M 1 .…”

“…As pointed in the literature (cf. [4]), explicitly knowing it can be useful specially when the function is not differentiable and you want to use an inequality like the one provided by the Mean Value Theorem. It is noteworthy that presenting such inequalities for a variety of spaces and functions is an endless endeavor in analysis; see for instance [1,5,6,9,13,12,14] and references therein.…”