Unified proofs of three fundamental properties of continuous functions

Daniel Daners
Preprint March 2020
Mathematics Magazine 95 (2022), 502–508.
Original article at doi:10.1080/0025570X.2022.2125725

Abstract

We provide a unified approach to three fundamental properties of continuous functions on closed and bounded intervals: the intermediate value theorem, the extreme value theorem and the uniform continuity. We prove all three using the same building block, only making use of the least upper bound axiom and the ε-δ definition of continuity.

A preprint is available on request.