Last time I posted some questions. Today I post the answer that I know.
- Is there a subset of [0,1] that is uncountable and has measure 0? YES- take the Cantor Set. Many readers knew this.
- Is there such a set that is natural? Some comments thought the Cantor Set was natural. I disagree, however
this is a matter of opinion and taste.
- Is there a function that is continuous everywhere and differential nowhere? YES- take the Weierstrass Function. Many readers knew this.
- Is there such a function that is natural? Some think the Weierstrass Function is natural. I disagree because it was constructed for the sole reason to be continuous everywhere and differential nowhere. But again, there are those who think it is natural and this is a matter of opinion.
- Is there a function that has derivative 0 at almost every points of [0,1] even though it is strictly increasing?
YES- take Cantor Functions,
also known as the Devil's staircase. Some readers knew this.
- Is there a natural such function? YES- and this was the motivation for the post. (I also didn't want to have the example I read about mentioned in the comments of the last post which is why I wanted all posts to be non-anonymous- so if I blocked one I could email the author WHY I blocked it.)
The function I will talk about came about because of a real problem having nothing to do with finding weird functions. Hence I think it is unambiguously natural.
I base my exposition on the exposition in
How to Gamble if you Must
by Kyle Siegrist. The original source is from the classic book Inequalities for Stochastic Processes: How to Gamble
if you must by Dubbins and Savage. Classic or not, unless that book is free online the future
will credit Siegrist with the result (even though Siegrist references Dubbins and Savage.)
Consider the following game: Let 0 < p < 0.5.
Let 0 ≤ x ≤ 1. We will view p as a probability and x as how
much money you start with.
You place a bet (which has to be ≤ what you have).
A coin is flipped with prob of WIN being p and of LOSS being 1-p.
If you win you double your bet. If you lose you lose your bet.
You repeat until you have 0 or you have 1.
You could play TIMIDLY: bet a fixed small amount every time.
This is analyzed. In the paper and is interesting.
You could play BOLDLY: if you have < 1/2 then bet all you have,
and if you have ≥ 1/2 then bet so that if you win you'll have 1
and can stop.
If you play BOLDLY then
what is the probability that you will end with 1?
We will fix p and let F(x) be the prob if you begin with x.
(We assume 0 ≤ x ≤ 1.)
F satisfies the following recurrence:
F(x) = pF(2x) if x ∈ [0,0.5]
F(x) = p + (1-p)F(2x-1) if x ∈ [0.5,1].
This is enough to define F on all rationals between 0 and 1 (use the expansion in base 2).
Then use continuity to define F on all the rationals between 0 and 1.
The paper claims that F has derivative 0 at almost every point of [0,1]
even though its strictly increasing.
I leave this for the readers (translation into English: I don't know the proof).
However, I emailed the author who emailed me the following:
There are proofs in the book Probability and Measure by Patrick
Billingsley. In the third edition, it's Example 31.1 on page 407. The fact
that F is continuous and strictly increasing on [0, 1] is fairly easy to
see, based on the definition of the underlying random variable. The fact
that F has derivative 0 is not as easy to see intuitively--the proof is
more of a hard analysis type proof.
The paper assumes that money is continuous and out lives are indefinitely long.
Ekhad, Georgiadis, Zeilberger have written a paper
How to Gamble if you're in a Hurry
that looks at these problems from (to quote their abstract) a purely discrete, finistic, and computational viewpoint.