# Cantor function

I have sometimes liked to learn math from textbooks by doing all, or most, of the exercises. This can be really engaging when it is going well.

A problem I have often run into is that I will get stuck. Sometimes talking about the problem unsticks it. Maybe writing about the problem will, too?

Currently the problem is the "Cantor Function" problem from chapter 6 of Understanding Analysis. Part (a) is easy. Part (b) though feels like it is just on the edge of sight. It feels like it "should" be simple once I figure out how to write it down. The general idea is: Once an interval $$[a, b]$$ is flattened out at say $$y_0 = f_N(x)$$ for $$x \in [a, b]$$, it stays flattened out for good: For all $$n \geq N$$ we have $$f_n(x) = f_N(x) = y_0$$. The plateaus "get closer together." Furthermore, the $$f_n$$s are increasing functions. We should be able to use these two facts to "force" the values to get closer together in a uniform convergence sort of way.

What I'd like to do is say: We can "trap" the values where the sequence of functions is not eventually constant. (Those should turn out to be the points of the Cantor set, but I don't feel like proving it.) Given some $$\epsilon > 0$$ I think we can force points to be close together by looking at the plateaus only. We only need an $$N$$ such that any consecutive pair of boundary points (that is 0 and 1) or plateaus is closer than $$\epsilon$$. Every function $$f_n$$ is increasing, and the plateaus in $$f_N$$ will remain in every $$f_n$$ with $$n \geq N$$. That means that if we can force the plateaus to be close enough in $$f_N$$, then for all $$n \geq N$$, the values of $$f_n$$ in between the plateaus/boundaries must be trapped between the $$f_N$$ values on the consecutive plateaus/boundaries.

This feels like it should work, but it sounds like a pain to formalize if you can do so at al. Unfortunately I expect the formalization to be important for figuring out (c). So... 🤷‍♂️

I suspect it might be possible to rescue this argument by finding an explicit formula for points inside the plateaus. I have my candidate points, namely those of the form $$k/2^n$$ where $$k, n \in \mathbb{Z_{\geq 0}}$$ and $$k \leq 2^n$$. In fact I think $$f_n$$ is constant at exactly the points $$k/2^n$$ with its value at such points being... $$k/2^n$$. This would be convenient, because if true, then it is obvious which $$N$$ will work: The distance between consecutive boundary points/plateaus would be $$1/2^N$$, so just take $$N$$ such that $$2^{-N} < \epsilon$$.

I'd have to prove that these candidate points work, though. That means writing down the recursion explicitly and probably doing an induction argument, which... eh. Maybe later.