Math205a Fall Term 2002

Assignment 8

1. Let A ⊂ ℝ be a closed set. Construct a function f ∈ C^{1}(ℝ, ℝ) such

that

f(x) = 0 ⟺ x ∈ A .

2. Let k ∈ ℕ and show that

f : (0, ∞) → ℝ , x ↦ x^{1/k}

can be defined as the inverse of g : (0, ∞) → ℝ , x ↦ x^{k} and compute f′.

3. Show that there exists a real real-valued continuously differentiable function which is invertible and for which the inverse is not Hölder continuous of any exponent.

4. Let f ∈ C^{2}((a, b), ℝ) and assume that f′′(x) ≥ 0 , x ∈ (a, b). Prove that

f(tx + (1 − t)y) ≤ tf(x) + (1 − t)f(y) , t ∈ [0, 1] , x, y ∈ (a, b) .

The Homework is due Wednesday December 4 2002