Volterra's Function - Context and History

Question

This is a rather soft question, but I am preparing a short presentation on Volterra's function for an analysis course. Specifically, I'm wondering why was the Volterra's function a big deal during it's day. I know a few years earlier Weierstrass found his function which served as a counter example to Ampere's proof that all continuous functions are differentiable.

I guess a more succinct way to ask my questions is as follows: how do I put Volterra's function into historical context? As of right now it seems to be viewed as something interesting and toy with, a nice intellectual exercise. However, when it was constructed, did it upset a lot of people? Was it controversial? So forth.

Answer 1

I think the level of dismay at so-called pathological functions was not excessive when in 1881 Volterra introduced his function with a non-integrable derivative. It may not have attracted much attention.

The level of dismay at this type of function may have risen when in 1895 Weierstrass's CND function was introduced/published. See, e.g., G.A. Edgar, Classics on Fractals, Addison-Wesley Publishing Company, 1993, 3–9. According to Korner (Fourier Analysis, p. 42), Lebesgue himself had trouble publishing an article that contained a CND function, suffering the objections of those who felt it muddied the waters of classical analysis. Note than Weierstrass's paper wasn't published until 1895, so the two ideas were--for all practical purposes--contemporaneous.

This paper does some of the work for you, but there isn't enough on CND functions to give historical perspective. If "pathological functions" illuminate the boundaries of classical analysis they should be examined together. Then it is easier to judge whether a particular function is a curiosity or a landmark. I think both (Volterra's and Weierstrass's functions) are more than curiosities.

This paper is also interesting.