Study the continuity of a function 
I have the function $f:\mathbb{R}^2\rightarrow\mathbb{R}\ f(x,y) = \left\{\begin{matrix}
\sin\frac{x^3y}{x^4+y^4}, & (x,y) \in \mathbb{R}^2 \setminus\{(0,0)\}\\ 
 0, & (x,y) = (0,0).
\end{matrix}\right.$ I need to study the continuity of the function $f$. 

I tried to calculate the limits as $(x,y) \rightarrow (0,0)$ of $f$. So $\lim_{(x,y)\rightarrow(0,0)}\sin\frac{x^3y}{x^4+y^4} = \sin(0) = 0$ So this would imply that $f$ is continuous at $(0,0)$, right?

I don't understand your solution. I'll give a hint on how to solve it. What limit do you get if you take $(x,y)\to (0,0)$ along the curve $(t,0)$? And how about the curve $(t,t)$? 
