Make a multivariable function continuous What can we do with this function, so the function will be continuous in $(0,0)$?
$f:\mathbb{R}^2\rightarrow\mathbb{R}:(x,y) \mapsto \frac{x^2+y^2-x^3y^3}{x^2+y^2}$
What I think we should do, is:
approximate $(0,0)$  via the line $y=x$, so substitute $y=x$ and take the limit of that function, i.e. $\lim_{x\rightarrow0}$:
$\lim_{x\rightarrow0} 1 - \frac{x^6}{2x^2} = \lim_{x\rightarrow0}1-\frac{x^4}{2} = 1$
So the new function, that is continuous in $(0,0)$ is defined by:
$F:\mathbb{R}^2\rightarrow\mathbb{R}:(x,y) \mapsto F(x) = \begin{cases} 1 & \mbox{if } (x,y) = (0,0) \\ f(x) & \mbox{if } (x,y) \neq (0,0)  \end{cases}$
I'm sorry for my english, but if you understand my question, could you say if I'm right? 

Yes, it is going to work, but you need to justify it better. 
In fact you only demonstrated that the limit along the line $y=x$ equals $1$, but to ensure $f$ continuos you have to be sure that the limit holds from whichever direction one takes. 
In particular, you have to set $x = \rho \cos \theta$, $y =  \rho \sin \theta$, and as you can see if $\rho \to 0$, then the $f \to 1$
