Past exam of the options II course of the University of Oxford / 2011 / F / F2 / b / iv

Solution

The partial derivatives satisfy a global bound of

1

in absolute value. For example,

∣ \partial_{x} f (x, y) ∣ \leq \frac{2∣ x ∣}{( 1 + x ^{2} ) ^{2}} \leq 1,

because

2∣ x ∣ \leq 1 + x^{2}

and

1 + x^{2} \geq 1

; the same argument applies to

\partial_{y} f

. Apply the one-dimensional mean value theorem first along a horizontal segment and then along a vertical segment:

∣ 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

f

is Lipschitz with constant

1

for the coordinate-sum norm.

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