In mathematics, a continuous nowhere differentiable function is a numerical function which is regular from the topological point of view (i.e. continuous) but not from the point of view of differential calculus (i.e. not differentiable at any point).
The continuity of a function means that its representative curve does not admit a "hole". The differentiation ensures that it is well "rounded". It is quite easy to demonstrate that any derivable function over an interval is continuous over this same interval. Mathematicians believed until the 19th century that the converse was partly true, that the points where a continuous function is not derivable are rare. It is not so. Many counterexamples were discovered.
Since then, the study of these functions has shown that they are important, not only from the point of view of the internal logic of mathematics, for understanding the concept of function, but also for providing useful models for other sciences. Fractals also give examples of continuous curves without tangents.
History edit
The very concept of function was only clarified in the 19th century, when in 1837 Dirichlet proposed a modern definition of the concept of function.
Une quantité y est une fonction (univoque) d'une quantité x, dans un intervalle donné quand à chaque valeur attribuée à x dans cet intervalle correspond une valeur unique et déterminée de y, sans rien spécifier sur la façon dont les diverses valeurs de y s'enchaînent les unes aux autres[1].
At that time, mathematicians thought that any continuous function is differentiable, except possibly at few specific points, this opinion not being contradicted by their practice of differential calculus [2]. For instance, in 1806, Ampère [3] tried to prove that any function is differentiable à l'exception de certaines valeurs particulières et isolées (with the exception of certain particular isolated values) without however clarifying what he meant by function [4].
From 1833-1834 [5], Bernard Bolzano presents the first example of a function that is continuous everywhere but nowhere differentiable by building the so-called Bolzano curve iteratively[6]. For him, a limit of continuous functions is a continuous function [7]. He shows that the function obtained is not monotonic in any interval and that it has no derivative in a dense set. This example is in fact richer because one can prove that its function has no derivative or even infinite derivative with a determined sign at any point except on the right of the origin where the ratio limit is +∞ [6]. But the manuscripts of his work on this function, called the Bolzano function, were not rediscovered until 1920 and published only in 1922 [8]. Charles Cellérier also discovered, around 1860, another example of a continuous nowhere differentiable function without knowing that of Bolzano. His work also remains unpublished until his death in 1890 [9].
This is why Bernhard Riemann surprised the mathematical community when he exhibited, at a conference in 1861, an example of a function which is continuous on but differentiable only in rare points [4]. This function is defined by
and is differentiable at x only when x = pπ/q where p and q are odd integers.
- ^ Encyclopédie des sciences mathématiques pures et appliquées, tome II, vol. 1, Gauthier-Villars, 1909, p. 13
- ^ Thim 2003, p. 4
- ^ André-Marie Ampère, "Recherche sur quelques points de la théorie des fonctions dérivées qui conduisent à une nouvelle démonstration du théorème de Taylor, et à l'expression finie des termes qu'on néglige lorsqu'on arrête cette série à un terme quelconque", Journal de l'École polytechnique, 13ème | cahier, tome VI, 1806, p. 148-191, [1]
- ^ a b Thim 2003
- ^ Jan Sebestik, Logique et mathématique chez Bernard Bolzano, Vrin, 1992, [2]
- ^ a b K. Rychlík, La théorie des fonctions de Bolzano. In Atti del Congresso Internazionale dei Matematici: Bologna del 3 al 10 de settembre di 1928, 1929, p. 503-506
- ^ This is not true in general, but in this case the convergence is uniform, so the continuity is indeed ensured.
- ^ Thim 2003, p. 11
- ^ Thim 2003, p. 18