Given , 

$$ f(x,y) = \begin{cases} 
 \dfrac{xy}{\sqrt{x^2+y^2}} & (x,y)\neq(0,0) \\
 0 & (x,y)=(0,0) \\ 
\end{cases} $$

We need to check whether the function is continuous at $(0,0)$ or not.

Applying the definition I get:

$\epsilon$ = $\sqrt\delta$.

but I do not know if I'm right