
Cantor Sets
Consider g(x)=x^2sin(1/x) if x>0 and 0 if x<=0
1. a) Find g'(0)
b) Compute g'(x) for x not 0
c)Explain why, for every delta>0, g'(x) attains every value between 1 and 1 as ranges over the set (delta,delta). Conclude that g'(x) is not continuous at x=0.
Next, we want to explore g with the Cantor set.
We have f_1(x)=0 if x is in [0,1/3]
=g(x1/3) if x is to right of 1/3
=g(x+2/3) if x is to left of 2/3
=0 if x is in [2/3,1]
Now f_2(x)=1/3f_1(3x) for x in [0,1/3]
=f_1(x) if x is in [1/3,2/3]
=1/3f_1(3x2) if x is in [2/3,1]
2. a) if c is in C(Cantor set) what is lim f_n(c)?
b) Why does lim f_n(x) exist for x not in C?
Now set f(x)=lim f(x)
3.a) Explain why f'(x) exists for all c not in C
b) If c in C, argue that f(x)<=(xc)^2 for all x in [0,1]. Show how this implies f'(c)=0
c) Give a careful argument for why f'(x) fails to be continuous on C.
4. Why is f'(x) Riemann integrable on [0,1]
The reason the Cantor set has measure zero is that, at each stage, 2^(n1) open intervals of length 1/3^n are removed from C_(n1). The resulting sum:
sum(2^(n1)(1/3^n) converges to 1, which means that C1, C2, C3,.... have total lengths tending to zero. Now let's remove intervals of lengths 1/3^(n+1)
5.Show that under these circumstances, the sum of the lengths of the intervals making up each C_n, no longer tends to zero as n tends to infinity. What is this limit?
I know this is way too many questions for one thread, but these are all directly related to each other.
I had already figured out 1, but left it there since it is used in the next problem.
2a) I think the answer is 0.
b) not quite sure
Any hints would help!