Homework 7, Advanced Calculus 1

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Homework 7, Advanced Calculus 1

1. Rudin Chapter 4 Exercise 20a.

2. Rudin Chapter 4 Exercise 20b.

3. Rudin Chapter 4 Exercise 21.

4. Rudin Chapter 4 Exercise 22.

The Cantor Function is defined by f : [0, 1] → [0, 1] with the following rules. Recall that every x ∈ [0, 1] can be written in tertiary expression x = P

ja_j3^−j, with a_j = 0, 1, 2.

The expression is unique except that

N −1

X

j=1

a_j3^−j+ a_N3^−N +

∞

X

j=N +1

2 · 3^−j =

N −1

X

j=1

a_j3^−j+ (a_N + 1)3^−N.

We pick the first expression to ensure uniqueness. The Cantor set C ⊂ [0, 1] is defined by those real numbers with a_j 6= 1 ∀j. We define f separately on C and C^c. For x =P

ja_j3^−j ∈ C, we define

f (x) =X

j

a_j 22^−j. For x =P

ja_j3^−j ∈ C^c, we define

f (x) =

Jx−1

X

j=1

a_j

22^−j+ 2^−Jx, where J_x is the first digit of x with a_j = 1.

Problems 5,6 concern the Cantor function and related topics.

5. Prove that the Cantor function f is uniformly continuous on [0, 1] and differentiable on C^c.

6. Prove that there exist constants C, α > 0 so that

|f (x) − f (y)| ≤ C|x − y|^α ∀x, y ∈ [0, 1].

7. Functions satisfying the condition in Problem 6 on its domain is said to be H¨older continuous with exponent α. Prove that

{H¨older continuous function} ( {Uniformly Continuous Functions}.

8. Rudin Chapter 4 Exercise 25.

9. Rudin Chapter 4 Exercise 26.

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