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I have to prove that this function

$f(x,y) = \frac{x^3y}{x^4+y^2}, \text{ if } (x,y) \neq (0,0)$

$f(x,y) = 0, \text{ if } (x,y) = (0,0)$

in the domain $D = \{ (x,y)\in\mathbb{R}^2:|x|\leq1, |y|\leq2 \}$.

Does not satisfy Lipschitz condition.

I'm using a theorem that says that Lipschitz condition $|f(x,y_1)-f(x,y_2)| \leq L|y_1-y_2|, \ \forall \ (x,y_1), (x,y_2) \in D $, is satisfied by $f(x,y)$ if and only if $\sup\limits_{(x,y)\in D} \left| \frac{\partial}{\partial y} f(x,y) \right| \leq L$.

I don't know how to justify properly that $\sup\limits_{(x,y)\in D} \left| \frac{\partial}{\partial y} f(x,y) \right|$ does not exist because $\frac{\partial}{\partial y} f(x,y)$ is not bounded.

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1 Answer 1

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If you compute $\frac{\partial f}{\partial y}(x,0)$ you get $\frac{1}{x}$, so you get the unboundness.

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