Is $f$ continuous at$(0,0)$?

Question

$f:\Bbb R^2\to \Bbb R$ given by $$ f(x, y) = \cases{\dfrac{2(x^3+y^3)}{x^2 + y} & if $(x, y)\neq (0,0)$\\ 0 & if $(x, y) = (0,0)$ } $$ It looks continuous, but my friend said it isn't. I tried to show discontinuity by taking various paths, but was only met with failure. I couldn't prove it is continuous either. Please help.

Answer 1

Note that for $x=t$ and $y=-t^2+t^3=-t^2(1-t)$ with $t\to 0$

$$\dfrac{2(x^3+y^3)}{x^2 + y}=2\cdot\dfrac{t^3-t^6(1-t)^3}{t^2-t^2+t^3}=2-t^3(1-t)\to 2$$

thus $f(x,y)$ in not continuos in $(0,0)$.

Answer 2

Sometimes it helps to do a plot:

Can you do it from here?