Another Proof of Darboux's Theorem

TL;DR

This paper presents a simple proof of Darboux's theorem, showing that derivatives on closed intervals have the Intermediate Value Property, using only the Intermediate Value Theorem and Rolle's theorem.

Contribution

It offers a new, elegant proof of Darboux's theorem that avoids reliance on the Extreme Value Property, simplifying the understanding of derivatives' behavior.

Findings

Derivatives on closed intervals satisfy the IVP property.

The proof uses only the Intermediate Value Theorem and Rolle's theorem.

The approach simplifies the existing proofs of Darboux's theorem.

Abstract

We know that a continuous function on a closed interval satisfies the Intermediate Value Property. Likewise, the derivative function of a differentiable function on a closed interval satisfies the IVP property which is known as the Darboux theorem in any real analysis course. Most of the proofs found in the literature use the Extreme Value Property of a continuous function. In this paper, I am going to present a simple and elegant proof of the Darboux theorem using the Intermediate Value Theorem and the Rolles theorem