Introduction to Analysis

Introduction to Analysis

Homework 6

1. (Rudin 5.15) Suppose a ∈ R, f is a twice-differentiable real function on (a, ∞), and M0, M1, M2 are the least upper bounds of |f (x)|, |f⁰(x)|, |f⁰⁰(x)|, respectively, on (a, ∞).

Prove that

M₁² ≤ 4M₀M₂. (There is hint in Rudin’s book. I skip it here.)

2. (Rudin 5.26) Suppose f is differentiable on [a, b], f (a) = 0, and there is a real number A such that |f⁰(x)| ≤ A|f (x)| on [a, b]. Prove that f (x) = 0 for all x ∈ [a, b]. Hint: Fix x₀ ∈ [a, b], let

M₀ = sup |f (x)|, M₁ = sup |f⁰(x)|

for a ≤ x ≤ x₀. For any such x,

|f (x)| ≤ M₁(x₀− a) ≤ A(x₀− a)M₀. Hence M0 = 0 if A(x0− a) < 1. That is , f = 0 on [a, x0]. Proceed.

3. (Rudin 5.27) Let φ be a real function defined on a rectangle R in the plane, given by a ≤ x ≤ b, α ≤ y ≤ β. A solution of the initial-value problem

y⁰ = φ(x, y), y(a) = c, (α ≤ c ≤ β)

is, by definition, a differentiable function f on [a, b] such that f (a) = c, α ≤ f (x) ≤ β, and

f⁰(x) = φ(x, f (x)) (a ≤ x ≤ b).

Prove that such a problem has at most one solution if there is a constant A such that

|φ(x, y₂) − φ(x, y₁)| ≤ A|y₂− y₁| whenever (x, y₁) ∈ R and (x, y₂) ∈ R.

Hint: Apply Problem 2 to the difference of two solutions.

4. (A mean-value theorem with possibly infinite end-points.)

Suppose −∞ ≤ a < b ≤ +∞, and f is differentiable function on (a, b) such that limx→af (x) = limx→bf (x). Show that there exists c ∈ (a, b) such that f⁰(c) = 0.

5. Let f : R → R be a differentiable function.

(a) Show that if f⁰ is bounded, then f is uniformly continuous.

(b) Show by example that the converse is not true.

(c) Show that if an example such as you are asked for in (b) is twice differentiable, then its second derivative must also be unbounded. Equivalently, (in view of (a)), show that a twice differentiable function R → R whose second derivative is bounded is uniformly continuous if and only if its first derivative is bound.

6. Let f : R → R be infinitely differentiable (meaning that f⁽ⁿ⁾) exists for all n ≥ 0), and suppose that for all n ≥ 0, lim_x→+∞f⁽ⁿ⁾(x) = lim_x→−∞f⁽ⁿ⁾(x) = 0. Show that for each n ≥ 0 there exist at least n distinct real numbers x such that f⁽ⁿ⁾(x) = 0.

7. (Definition) Let D ⊂ R^N and suppose f : D → R is a given function. We say that f is homogeneous of degree n if and only if (i)kx ∈ D whenever x ∈ D and k ∈ R − {0}, and (ii)

f (kx) = kⁿf (x) for x ∈ D, k 6= 0.

We also say that f is positively homogeneous of degree n if and only if the two conditions above hold for all k > 0 and all x ∈ D.

Prove the Euler’s theorem on homogeneous functions: Suppose that f : R^N → R is positively homogeneous of degree n and suppose that f_,1, f_,2, · · · , f_,N are continuous for a 6= 0. Then

N

X

i=1

a_if_,i(a) = nf (a) where a = (a₁, a₂, · · · , a_N).