Mean Value Theorems for Vector Valued Functions

Abstract

The object of this paper is to give a generalisation to vector valued functions of the classical mean value theorem of differential calculus. In that theorem we havefor some c in the open interval a, b when f is a real valued function which is continuous on the closed interval a, b and differentiable on the open interval. The counterpart to (1) when f has values in an n-dimensional vector space turns out to bewhere cka, b, 0 k, and .