Father Xmas visits real function land

Abstract:

Lots of theorems in ?mathematical methods? only apply to sufficiently ?nice? functions. This is OK for applied maths as ?Nature is not as malicious as Tripos examiners? (attributed to a Cambridge Mathematical Physics lecturer); but just how many functions actually are ?nice?? In this talk I will show that not only can functions get pretty much arbitrarily nasty, but in a certain technical sense first studied by Rene Baire in 1899, there are far more nasty functions than nice functions.

In the talk, I define Baire classes. The 0th class consists of continuous functions. The nth class consists of limits of convergent sequences of (n-1)th class functions. I then pose (and answer!) three questions:

? What do first class functions look like?

? How many distinct classes are there?

? Are there any functions so bad, they?re not in any Baire class?

Here are the slides I used for the talk (as a Word file).
Here's an animation illustrating how to grow piece-wise continuous functions out of continuous ones (as a Maple 6/7 file).

My bibliography is:

Books:

? Baire, R. Sur les functions des variables réelles. Ann. Mat. Pura. Appl. 1899. - where it all began.
? Goffman, C. Real Functions. Prindle. Weber and Schmidt. 1952 ? a more modern perspective
? Wier, A. J. Lebesgue Integration & Measure. Cambridge. CUP. 1973. ? a very brief summary
? Stillwell, J. Mathematics and its History. New York. Springer-Verlag. 1989. ? for infinite arithmetic.

Website:

Biographical details

---

Back