On the Intermediate Point Principle

Abstract

In the Euclidean space of all real numbers, we know that the Brouwer fixed point theorem and Poincaré–Miranda’s theorem are derived directly from the intermediate value theorem. So, Brouwer’s theorem is a partial extension of the intermediate value theorem. Then, in this note, we prove an extension of the intermediate value theorem itself in the n-dimensional Euclidean space. Maybe it should be called the intermediate point principle.