M2: Analysis III

Question 1

Theorem

Continuous Functions are Integrable

Answer 1

Let f: [a, b] → ℝ be a continuous function.
Then, f is integrable.

Question 2

Proof

(Continuous Functions are Integrable)
Let f: [a, b] → ℝ be a continuous function.
Then, f is integrable.

4 points

Answer 2

• f is uniformly continuous
• Take partition with mesh < δ and optimal majorant / minorant
• f attains its bounds on each interval by Boundedness Theorem
• I(Φ₊) - I(Φ₋) < ε (b - a)

Question 3

Theorem

Bounded continuous functions are integrable

Answer 3

Let f: (a, b) → ℝ be a bounded continuous function.
Then, f is integrable.

Question 4

Proof

(Bounded Continuous Functions are Integrable)
Let f: (a, b) → ℝ be a bounded continuous function.
Then, f is integrable.

4 points

Answer 4

• f is uniformly continuous on [a + ε, b - ε] and bounded by M
• Take partition x₀ = a, x₁ = a + ε, xₙ₋₁ = b - ε, xₙ = b and mesh < δ
• Take optimal majorant / minorant with value M on (x₀, x₁)
• f attains its bounds on each non-endpoint interval by the Boundedness Theorem

Question 5

Proof

Suppose that f: [a, b] → ℝ is continuous. Then there is some c ∈ [a, b] such that ∫ₐᵇ f = (b - a) f(c).

3 points

Answer 5

• f attains its maximum M and minimum m by the Boundedness Theorem
• m ≤ (1 / (b - a)) ∫ₐᵇ f ≤ M
• Apply the Intermediate Value Theorem

Question 6

Proof

Suppose that f: [a, b] → ℝ is continuous, and that w: [a, b] → ℝ is a nonnegative integrable function. Then there is some c ∈ [a, b] such that ∫ₐᵇ f w = f(c) ∫ₐᵇ w.

3 points

Answer 6

• f attains its maximum M and minimum m by the Boundedness Theorem
• m ≤ (∫ₐᵇ f w) / ∫ₐᵇ w ≤ M if ∫ₐᵇ w ≠ 0
• Apply Intermediate Value Theorem

Question 7

Theorem

Monotone Functions are Integrable

Answer 7

Let f: [a, b] → ℝ be a monotone function.
Then, f is integrable.

Question 8

Proof

(Monotone Functions are Integrable)
Let f: [a, b] → ℝ be a monotone function.
Then, f is integrable.

3 points

Answer 8

• Partition [a, b] into n equal parts
• Φ₊(x) = f(xᵢ) on (xᵢ₋₁, xᵢ)
• I(Φ₊) - I(Φ₋) = (1/n) (b - a) (f(b) - f(a))

Question 9

Theorem

First Fundamental Theorem of Calculus

Answer 9

Suppose that f is integrable on (a, b). Define a new function F: [a, b] → ℝ by F(x) := ∫ₐˣ f.
Then F is continuous. Moreover, if f is continuous at c ∈ (a, b) then F is differentiable at c and F’(c) = f(c).

Question 10

Proof

(First Fundamental Theorem of Calculus)
Suppose that f is integrable on (a, b). Define a new function F: [a, b] → ℝ by F(x) := ∫ₐˣ f.
Then F is continuous. Moreover, if f is continuous at c ∈ (a, b) then F is differentiable at c and F’(c) = f(c).

3 points

Answer 10

• |F(c + h) - F(c)| ≤ M h
• F is uniformly continuous taking δ = ε / M
• |F(c + h) - F(c) - h f(c)| ≤ ε h for any h ∈ (0, δ)

Question 11

Theorem

Second Fundamental Theorem of Calculus

Answer 11

Suppose that F: [a, b] → ℝ is continuous on [a, b] and differentiable on (a, b). Suppose furthermore that its derivative F’ is integrable on (a, b).
Then ∫ₐᵇ F’ = F(b) - F(a).

Question 12

Proof

(Second Fundamental Theorem of Calculus)
Suppose that F: [a, b] → ℝ is continuous on [a, b] and differentiable on (a, b). Suppose furthermore that its derivative F’ is integrable on (a, b).
Then ∫ₐᵇ F’ = F(b) - F(a).

2 points

Answer 12

• We show some Riemann sum Σ(F’; P; ξ) = F(b) - F(a)
• F’(ξᵢ) (xᵢ - xᵢ₋₁) = F(xᵢ) - F(xᵢ₋₁) for some ξᵢ ∈ (xᵢ₋₁, xᵢ) by the Mean Value Theorem

Question 13

Proof

Suppose that fₙ: [a, b] → ℝ are integrable, and that fₙ → f uniformly on [a, b]. Then f is also integrable, and ∫ₐᵇ fₙ → ∫ₐᵇ f as n → ∞.

4 points

Answer 13

• |fₙ(x) - f(x)| ≤ ε
• I(Φ₊) - I(Φ₋) ≤ ε
• ψ₊ := Φ₊ + ε
• |∫ₐᵇ fₙ - ∫ₐᵇ f| ≤ sup{|fₙ(x) - f(x)|: x ∈ [a, b]} → 0