The Mean Value Theorem in the Context of Generalized Approach to Differentiability

Abstract

The article is a natural continuation of the systematic research of the properties of the generalized concept of differentiability for functions with a domain X⊂Rn that is not necessarily open, at points that allow a neighbourhood ray in the domain. In the new context, the well-known Lagrange’s mean value theorem for scalar functions is stated and proved, even for the case when the differential is not unique at all points of the observed segment in the domain. Likewise, it has been proven that its variant is valid for vector functions as well. Additionally, the paper provides a proof of the generalization of the mean value theorem for continuous scalar functions continuously differentiable in the interior of a compact domain.

Keywords

• the mean value theorem
• generalized approach to differentiability
• neighbourhood ray
• linearization space