Continuity of a complicated function with four parameters

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.

Use polar coordinates: $$y=r\sin{t}$$ $$x=r\cos{t}$$ $$r=\sqrt{x^2+y^2}$$ where $$t\in [0,2\pi)$$
• Oh I see, to put this in an easier way, I should make two cases : $p>q$ and $q \geq p.$ – user700974 Oct 30 '19 at 13:32