MATH 321:201: Real Variables II (Term 2, 2010)
Home work assignment #4
Due date: Friday, Feb. 5, 2010 (hand-in in class)
Problem 1 : Do [Rudin, Ch7. Exercise #11].
Problem 2: Do [Rudin, Ch7. Exercise #12].
Problem 3: Do [Rudin, Ch7. Exercise # 14].
Problem 4: Let X, Y be metric space with metrics dX , dY , respectively. Let E ⊂ X. A function
f : X → Y is said to be Lipschitz on E if there exists a constant C > 0 such that for all x, y ∈ E
dY (f (x), f (y)) ≤ CdX (x, y)
Here, for a given function f , the infimum of such constants C is called Lipschitz constant of f . (It
depends on E.) Consider a sequence of functions fn , f : X → Y , n = 1, 2, 3, · · · .
(a) Define uniform convergence of fn on E.
(b) Suppose fn → f uniformly on E. Assume fn are Lipschitz on E with the Lipschitz constant
Cn . Assume there exists C > 0 such that Cn ≤ C for all n ≥ 1. Is f also Lipschitz on E? If so,
then what can you say about its Lipschitz constant?
(c) In (b), remove the assumption that Cn ≤ C, and answer the same question for f ?
The following are suggested exercises. Please DO NOT hand-in, but, it is important for you to
do these suggested exercises!
Problem: Do Rudin, Ch. 7, Exercises # 10, #13.
Problem: Let α : [a, b] → R be a monotonically increasing function. Assume the sequence of
functions fn ∈ R(α) on [a, b] converges to f ∈ R(α) in L1 on [a, b], i.e.
Z b
|fn − f |dα = 0.
(∗) · · · lim
n→∞
a
Then, is it true
Z
(∗∗) · · ·
b
Z
f dα = lim
a
n→∞
b
fn dα?
a
Is the converse true? Namely, if (∗∗) holds, then does (∗) hold?
Problem: Study [Rudin, Theorem 7.18] and do the following: Consider instead
∞
X
2 n
g(x) =
ϕ(4n x),
4
n=0
where ϕ : R → R is defined as
ϕ(x) = |x| (−1 ≤ x ≤ 1)
ϕ(x + 2) = ϕ(x).
(a) What can we say about the differentiability of g?
(b) Consider more generally
gα (x) =
∞
X
α n
ϕ(4n x),
4
n=0
for a positive real number α > 0. The example in [Rudin, Theorem 7.18] is the case α = 3. What
can we say about differentiability of gα ? Are there different ranges of α where gα shows different
behaviors regarding differentiability?
2