$f (x,y)=\frac {x^2y}{x^3-y^2} $ is not continuous

Question

Show that $f:\mathbb {R}^2\rightarrow \mathbb {R} $, $f (x,y)=\frac {x^2y}{x^3-y^2} $ if $x^3\neq y^2$ and $0$ otherwise.

I have to prove that $f$ is not continuous.

I try to prove with sequence. If $x_n=y_n=\frac {1}{n} $ then the limit is 0. I need another sequences.

Answer 1

If $x = y$, then $f(x,y) = f(x,x) = \frac{x^3}{x^3-x^2}.$ Observe that $\lim_{x\to 1} f(x,x) = \pm\infty$, yet $f(1,1) = 0$.

If you want to use sequences, take $x_n = y_n = 1+ \frac 1n$. In that case $f(x_n,y_n) = 1 + n$.

Answer 2

Note that if $x=\dfrac1{n^2}+\dfrac1{n^3}$ and $y=\dfrac1{n^3}$, then$$f(x,y)=\frac{(1+n)^2}{3n^2+3n+1}\to\frac13.$$