ABSTRACT
Fourier's realization that one can represent even discontinuous functions as sums of sines and cosines forced mathematicians, sometimes against their will, to leave the security of the clearing to venture into the forest. There they found bizarre functions, like the one Weierstrass (1 8 15-1 897) showed "to his astounded and often indignant contemporaries" [Meyer 5, p. 121, a function everywhere continuous and nowhere differentiable, which fluctuates incessantly and whose graph is fractal, each part containing "the same complexity as the whole of the graph":
"I turn with terror and horror from this lamentable scourge of continuous functions with no derivatives.. . ," wrote Charles Hennite in 1893 to his friend the geometer Thomas Stieltjes [Hermite, p. 3 181. But for those who mourned the honest functions of their fathers, there was worse to come. The notion of a function was to be profoundly transformed by the general definition of an integral by Lebesgue (1875-1941), the creation of function spaces (Stefan Banach, Felix Hausdorff, . . . ) and the theory of distributions (Laurent Schwartz, Israel Gelfand).
