Physicist Amok

Everyone is familiar with the derivative of a function in terms of limits: for a sequence converging to , . I spent a couple days playing with sequences which accumulate but do not converge, seeing if I could do calculus without limits. I came to my senses and realized I’m a biologist, but not before I stumbled across this:

Treat values of a sequence as values of a random variable with uniform probability density. Then if has an accumulation point at , . To see this, when you’re close to , you get enormous denominators. Since you get arbitrarily close to arbitrarily often, you have infinitely many denominators as large as you like. These completely swamp any contribution of points of the sequence away from .

I suspect that there is a “fundamental theorem of analysis” which says that a statement about a space is true is equivalent to the statement being true at the accumulation points of all accumulating sequences in that space. But I don’t know how to define the above expectation except as a limit of finite sequences, so this doesn’t advance the program at all.

(Before people misunderstand, I like limits. I use them constantly. Some of my best friends are limits. This is a mathematical diversion.)