Can a Discontinuous Function Be Imposed on a Continuous Structure?

[Bob3141592]

I have a question about the Dirichlet function, which is 1 at the rationals and 0 at the irrationals. I had thought this was an "arbitrary" definition, but then I saw the function given algebraically as f(x) = \lim_{k\rightarrow\infty}\left( \lim_{n\rightarrow\infty}\left( Cos( k! \pi x ) ^{2n} ) ) I understand how rational x is p/q and eventually the k! will multiply out the denominator leaving an even multiple of pi in the cosine for 1. An irrational x always produces a cosine less than 1, which the even exponent then drives to a strictly positive zero. Each point converges, but it's nowhere continuous. That seems clear enough, but still there's something about how all this works that I find disturbing.

If we were to modify this equation just a little bit, by substituting \Gamma(k) instead of the factorial, we have a completely different situation. The limit no longer exists. There's a similar issue with the exponent n, which has to be integral because the cosine is negative half the time. Yet the second function contains the first. Especially as they approach infinity, why should this make any difference?

The more I think about it the more mixed up I get. Are there any rules about how to handle these mixing of cardinalities?

 

[lurflurf]

It is important and implied that n and k are natural numbers. If you take them real of complex the limit will not exist.

 

[Bob3141592]

Yes, but then I guess my question is why is it important? I'd always thought the Dirichlet function said something about the real numbers, but lately I'm thinking it doesn't. The restricting of k to integers is a narrow filter, and it's like the difference in cardinalities is an impedance mismatch. You can't make a clear reading that way. It's like trying to map the ocean by only looking at the tops of the waves. The sampling error can eliminate the entire signal.

Perhaps my unease is because the function doesn't say anything about the real numbers, but more about how limits work. Does this function only mean that I can impose a discontinuous facade on a continuous structure if I sample a negligible number of points? Is that correct? And would that imply that when I take the limit to a point, there are attributes of that point that can be different if I take the limit in a different way?

If no limit exists when \Gamma(k) varies continuously, then I haven't distinguished between rationals and irrationals in the reals. That distinction is an artifact of the restrictions in the limit process chosen. Well, that may be overstating things, since I'm not sure I understand what I'm trying to say. Something to do with mixing of cardinalities bringing nothing but trouble, at least to me.

 

[Ben Niehoff]

You could also construct something like

\prod_{k=1}^{\infty} \sin^2 (k \pi x) \right

I'm too lazy to put in convergence factors, but it could be done. If x is rational, the product must terminate (and is equal to 0). Otherwise, it never terminates, and I think with care you could make it equal to 1 (by inserting something appropriate).

 

[lurflurf]

Another fun one is g=lim n*sin(2pi*x*n!).
In particular
g(e)=2pi
g(rational number)=0

In calculus one likes continuous functions, or at least almost continuous. If one wants to tell rational and irrational numbers apart one need a very discontinuous function.


"Does this function only mean that I can impose a discontinuous facade on a continuous structure if I sample a negligible number of points?"
Would a continuous facade on a discontinuous structure be less objectionable? We are defining a discontinous function, the discontinuity has to come from somewhere.
"Is that correct?"
yes
"And would that imply that when I take the limit to a point, there are attributes of that point that can be different if I take the limit in a different way?"
No, the other limit would fail in its purpose. Even if the limit were used to define the property, using a different limit would define a different version of the property. Like in integration we have many versions of integrable depending which limit we use.
"If no limit exists when varies continuously, then I haven't distinguished between rationals and irrationals in the reals. "
The function is well defined. It distinguished between rationals and irrationals in the reals. That limit is just a particular way to compute the function. If you do not like
lim (1+x/n)^n
it does not mean you dislike exp(x), it can be repressented differently.
"That distinction is an artifact of the restrictions in the limit process chosen. "
Limit processes are often restricted. Sums and products are often taken over integers. Every limit comes with a set of permissible directions, just as every function comes with a set of permissible arguments.