Monsters in the mathematics classroom: Learning analysis through the works of Gaston Darboux

International audienceThe drama of the rise of rigor in nineteenth century mathematical analysis has now been widely rehearsed. Notable within this saga is the appearance of functions with features so unexpected (e.g., everywhere continuous but nowhere differentiable) that contemporary critics described them as “bizarre”, “pathological,” or even “monsters.” Among the “monster-makers,” one of the most influential was Gaston Darboux (1842-1917). This paper reviews Darboux's mathematical and “backstage” contributions to the development of nineteenth century analysis, including some of his own favorite pet monsters, and explores the important role played by these “pathological functions” in the historical re-shaping of analysis. We then consider how these functions can be used in to help students develop a more robust understanding of modern analysis. We examine in particular how Darboux's proof of the result now known as “Darboux's Theorem” (i.e., all derivatives have the intermediate value property) in his 1875 Mémoire sur les fonctions discontinues can be used in today's analysis classroom.