Fonction constante et d\'eriv\'ee nulle : un r\'esultat si trivial..

TL;DR

This paper examines different proofs of the theorem that a differentiable function with zero derivative on an interval is constant, exploring their educational and historical significance in French secondary and early undergraduate curricula.

Contribution

It analyzes the relationships between various proofs of the theorem and their pedagogical, epistemological, and historical contexts in French education.

Findings

Different proofs have distinct pedagogical and historical implications.

The theorem's proof complexity varies with educational level.

Insights into curriculum development for teaching calculus.

Abstract

We study various proofs of the caracterization of constant functions, more precisely of the theorem: a derivable function, defined on a real interval, is constant if, and only if, its derivative is null. Our aim is to study the relationships of these proofs with the mathematical curriculum of secondary schools and the begining of undergraduate studies in France, from various point of views (epistemological, historical, didactical).