Is this function continuous at (0,0) I am trying to solve whether $f(x,y) = \frac{xy}{(x^2+y)}$ is continuous at$(0,0)$ or not where $f(0,0)$ is defined to be $0$. I converted this to polar by substituting $x=r\cos p,  y=r\sin p$ where $r$ tends to $0$ and got the result it is continuous. Book says it is not continuous. Where am I wrong and how to solve it?

Let $x=t$ and $y=-t^2+h(t)$, where $h(t) \to 0$ as $t\to 0$, define a parametric curve.  Then, we see that on this curve
$$\begin{align}
f(x,y)&=\frac{xy}{x^2+y}\\\\
&=\frac{t(-t^2+h(t))}{t^2+(-t^2+h(t))}\\\\
&=\frac{t}{h(t)}(h(t)-t^2)\\\\
\end{align}$$
Now, if $h(t)=t^4$, for example, then we see that along this parametric curve 
$$\begin{align}
\lim_{(x,y)\to (0,0)}f(x,y)&=\lim_{t\to 0}f(t,-t^2+t^4)\\\\
&=\lim_{t\to 0^{\pm}}\left(t-t^{-1}\right)\\\\
&=\mp \infty
\end{align}$$
and therefore, the function $f(x,y)$ cannot be continuous at $(0,0)$.
