 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2013/f/1/b/ii/solution

For every

z \in B

z_{3} + 3 > 2

and

∣ z_{1} + z_{2} ∣ < 2

. Consequently each of the three partial derivatives in the preceding gradient has absolute value below

1/2

, so

∥\nabla g (z) ∥^{2} < \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \frac{3}{4} .

The segment joining

x, y \in B

lies inside

B

∣ g (y) - g (x) ∣ \leq ∥\nabla g (x + s (y - x)) ∥ ∥ y - x ∥ \leq \frac{3}{2} ∥ y - x ∥ .

For

x = y

both sides vanish. This proves the required Lipschitz continuity uniformly on the ball.

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