The Intermediate Value Theorem and Its Applications

Abstract

The Intermediate Value Theorem (IVT) is a fundamental result in real analysis that ensures the existence of intermediate values for continuous functions defined on closed intervals. This theorem states that if a function is continuous on a closed interval and takes different values at the endpoints, it must take every value between those two endpoints at least once. The IVT has important applications in mathematics, including root-finding methods like the bisection method, and in proving the existence of solutions to equations and differential systems. Beyond mathematics, it is widely used in fields such as economics, physics, and engineering to model continuous systems and predict equilibrium points. This article explores the formal statement, proof, and key applications of the IVT, illustrating its broad relevance in both theoretical and applied contexts.