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Let $k,l,p,q \in \mathbb{N}$ be four positive integers and let $f$ be the function defined by $$ f(x,y) = \left\{ \begin{array}{ll} \frac{x^k y^l}{x^{2p} + y^{2q}} & \mbox{if } (x,y) \neq (0,0) \\ 0 & \mbox{if } (x,y) = (0,0) \end{array} \right. $$ Show that $f$ is continuous at $(0,0)$ if and only if $$\frac{k}{p}+\frac{l}{q} > 2.$$

I have some trouble to find the good inequalities for the "if" part and the good sequences for the "only if" part.

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ΗΙΝΤ

Use polar coordinates: $$y=r\sin{t}$$ $$x=r\cos{t}$$ $$r=\sqrt{x^2+y^2}$$ where $t\in [0,2\pi)$

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  • $\begingroup$ Shouldn't the interval be a half-closed interval? $\endgroup$ – Botond Oct 30 '19 at 13:23
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    $\begingroup$ @Botond yes..i edited.thank you for the observation $\endgroup$ – Marios Gretsas Oct 30 '19 at 13:24
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    $\begingroup$ Oh I see, to put this in an easier way, I should make two cases : $p>q$ and $q \geq p.$ $\endgroup$ – user700974 Oct 30 '19 at 13:32

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