 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2011/f/f2/b/iv/solution

Because

2∣ x ∣ \leq 1 + x^{2} \leq Q

, we have

∣ f_{x} (x, y) ∣ = 2∣ x ∣/ Q^{2} \leq 1/ Q \leq 1

; similarly

∣ f_{y} ∣ \leq 1

. Apply the one-variable mean value theorem on the horizontal and vertical segments joining

(x, y)

(u, y)

(u, v)

∣ f (x, y) - f (u, v) ∣ \leq ∣ f (x, y) - f (u, y) ∣ + ∣ f (u, y) - f (u, v) ∣ \leq ∣ x - u ∣ + ∣ y - v ∣ .

Thus the function is globally one-Lipschitz for the

ℓ^{1}

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