The Intermediate Value Theorem for Linear Transformations

Abstract

If a real-valued function is continuous on a real interval and it takes on two different values, then it will also take any value in between those two, by the Intermediate Value Theorem. It is not immediately clear what would be a natural generalization for functions whose domain and range are in higher-dimensional Euclidean spaces. In this article, we analyze this problem, by first arriving at what we think is the appropriate question to ask, and then restricting to linear transformations. It turns out that the matrices that will satisfy an appropriate version of the Intermediate Value Theorem are the so called {\em monotone} and {\em weakly monotone} matrices, which have applications in numerical approximation of the solutions to systems of linear equations.