Math205b Winter Term 2003

Assignment 11

1. Let a < c < b ∈ ℝ and f ∈ R([a, b], ℝ). Show that

f ∈ R([a, c], ℝ) ∩ R([c, b], ℝ) and
∫_{a}^{b} f(x) dx = ∫_{a}^{c} f(x) dx + ∫_{c}^{b} f(x) dx .

2. Let a < b ∈ ℝ and f ∈ R([a, b], ℝ). Define F : [a, b] → ℝ by

F (x) = ∫_{a}^{x} f(y) dy , x ∈ [a, b] .

Prove that F ∈ C^{1−}([a, b], ℝ). When is F differentiable? Let (x_{n})_{n∈ℕ} be a sequence in ℝ^{m} for some m ≥ 1. Then

x_{n} ⟶_{n→∞} x_{∞} ∈ ℝ^{m} : ⟺

[∑_{k=1}^{m}(x_{n}^{k} − x_{∞}^{k})^{2}]^{\frac{1}{2}} =: |x_{n} − x_{∞}|_{2} ⟶_{n→∞} 0

A function f : [a, b] → ℝ^{n} is said to be Riemann-integrable, or concisely f ∈ R([a, b], ℝ^{n}) iff

lim_{△(P)→0} S(f, P) ∈ ℝ^{n}

exists for every Cauchy sum.

3. Show that f ∈ R([a, b], ℝ^{n}) ⟺ f^{k} ∈ R([a, b], ℝ) , k = 1, …, n

4. Let the function f : [0, 1] → ℂ be given by

f(x) = \begin{cases}xe^{2iπ/x} , & x ≠ 0 \\ 0 & x = 0 . \end{cases}

Show that f ∈ C([0, 1], ℂ) and plot f([0, 1]).

5. You ask a question.

The Homework is due on Friday January 31