Show that the function is not continuous using the sequence definition. Consider $f$, given by $f(x,y)$= $xy^2/(x^2+y^4)$ when $(x,y)\ne(0,0)$ and $f(0,0)=0$.

Show that $f$ is not continuous at $(0,0)$ using the sequence definition. (which is: a function is continuous if any sequence of points $(X_n,Y_n)$ in $R^2$ with $\lim X_n=0$  and  $\lim Y_n=0$  then  $\lim f(X_n,Y_n)=f(0,0)=0$.



****Sorry, I don't know how to type piecewise functions****
Take the sequence $(x_n,y_n)  = (\frac{1}{n^2}, \frac{1}{n} ) \to (0,0) $. If $f$ were continuous at $(0,0)$, then we must have (by def) that $f( x_n,y_n) \to f(0,0) = 0 $. But,

$$ f(x_n,y_n) = \frac{\frac{1}{n^2} \cdot \frac{1}{n^2}}{\frac{1}{n^4} + \frac{1}{n^4}} = \frac{\frac{1}{n^4}}{\frac{2}{n^4}} = \frac{1}{2} \neq 0.$$
Choose $a_n=(\frac{1}{n},0)$ and $b_n=(\frac{1}{n^2},\frac{1}{n})$