PROBLEM SET 3 (DUE IN LECTURE ON OCT 1 (THURSDAY))
(All Theorem and Exercise numbers are references to the textbook by Apostol; for instance
“Exercise 1.15-3” means Exercise 3 in section 1.15.)
Problem 1. Do Exercise 1.15-3.
Problem 2. Do part (a) of Exercise 1.15-5.
Problem 3. Do Exercise 1.26-20.
Problem 4. Compute the integral
Z 1
1 + x + x2
dx.
1 + x2
−1
Problem 5. The characteristic function (also called the indicator function) of a subset
A ⊆ R is the function χA : R → R defined by
(
1 if x ∈ A
χA (x) :=
0 if x ∈
/ A.
Prove that χQ is not integrable on the unit interval [0, 1]. (Recall that Q is
the set of rational numbers. You may use the results of Exercises I 3.12-6 and
I 3.12-9 without proof.)
Problem 6. Let S0 = [0, 1] be the closed unit interval and let Si+1 be defined for each
i ≥ 0 by removing the open middle third of each interval in Si . Thus S1 =
[0, 13 ] ∪ [ 23 , 1], S2 = [0, 91 ] ∪ [ 92 , 13 ] ∪ [ 23 , 49 ] ∪ [ 89 , 1], and so on; in general Si is the
union of 2i closed intervals of length 3−i . The Cantor set C is defined as the
intersection of all the Si :
\
C :=
Si | i ≥ 0 .
R1
Prove that 0 χC (x)dx = 0, where χC is the characteristic function (as defined
in the previous problem) of the Cantor set.
1