Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is a fundamental theorem in calculus that states that if a continuous function takes on two distinct values, it must also take on every value between those two values within the given interval. In other words, if a continuous function passes through two points on a graph, the function must pass through every y-value between those points as well.

The IVT is often used to prove the existence of roots for continuous functions on a closed interval. The theorem is based on the properties of continuous functions, which are functions without any breaks or jumps in their graphs. The IVT has applications in various areas of mathematics, including the study of differential equations and numerical analysis.