The complexity of nowhere differentiable continuous functions

T. I. Ramsamujh
1989 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
Introduction. It was not always clear that there could exist a continuous function which was differentiable at no point. (Such functions are now known as nowhere differentiable continuous functions. By "differentiable" we mean having a finite derivative.) In fact in 1806 M. Ampere [2] even tried to show that no such function could exist but his reasonings were later discovered to be fallacious. Of the early attempts at constructing a nowhere differentiable continuous function mention must be
more » ... mention must be made of B. Bolzano. In a manuscript dated around 1830, (see [21] ) he constructed a continuous function on an interval and showed that it was not differentiable on a dense set of points. (It was later shown by K. Rychlik [21] that this function was in fact nowhere differentiable.) Around 1873 K. Weierstrass gave the first legitimate example of a nowhere differentiable continuous function. This discovery was published by Du Bois-Reymond [6] and prior to this no such function was ever published. Another example published in 1890 (see [5] ) was thought to have been discovered by C. Cellerier as early as 1850 but of this there is much doubt. Also a function studied by B. Riemann around 1860 and very often thought of as being nowhere differentiable turned out to be differentiable at certain points (see [7], [8] or [25] ). So Weierstrass holds sole claim for the first discovery. Later many more examples of nowhere differentiable continuous functions were constructed and it became fashionable to ask that more stringent requirements be satisfied (for instance, instead of being nowhere differentiable, the function might be required to have no derivative, finite or infinite). In 1925 A. Besicovitch [4] constructed a continuous function with no one-sided derivative, finite or infinite. Such functions are called Besicovitch functions in honour of their discoverer. There was, however, much controversy about Besicovitch's example because the construction was rather complicated and the reasoning could not be readily followed. E. D. Pepper [19] later examined the same example but there were still doubts in the minds of some as to the existence of such functions. These doubts were put to rest by A. P. Morse [17] who gave an example which satisfied even more stringent requirements than those of Besicovitch functions. The studies on nowhere differentiable continuous functions took a different twist when in 1931 S. Mazurkiewicz [16] showed that the set of all such functions is a comeager subset of the set of all continuous functions of period 1. At about the same time Banach [3] found the same result except that in this case the functions were defined on the set [0,1]. So in the sense of Baire Category the continuous functions which are not nowhere differentiable are exceptional. This
doi:10.4153/cjm-1989-004-9 fatcat:sexmur2pjzetdhid7qxjvosb3m