# The continuity of a function of two variables

How can I demonstrate the continuity of this function. If $a>0$, \begin{align} f(x,y) = \left\{ \begin{array}{lll} \dfrac{x^3y}{ax^4+y^4} & \text{if} & (x,y) \ne (0,0) \\ 0 & \text{if} & (x,y) = (0,0) \end{array} \right. \end{align}

I suppose I'm gonna have to calculate the limit of the function when $(x,y) \rightarrow (0,0)$. But I don't understand what that's gonna bring to my proof.

How can I continue my proof?

Approach to the point $(0,0)$ along the path $y=mx$.
Which shows that $f$ is discontinuous at $(0,0)$.
• Can you show how to calculate this limit? I suppose $$lim \frac{x^3(mx)}{ax^4+(mx)^4}=\frac{mx^4}{ax^4+m^4x^4}=\frac{m}{a+m}$$. – hlapointe Oct 20 '15 at 17:31
• That limit depends on $m$ ..and so $f$ is discontinuous at $(0,0)$ – Empty Oct 20 '15 at 17:32