Is this function continuous at $(0, 0)$? Suppose a function $f(x, y)$ is defined as follows like this: $f(x, y)=\frac{xy^3}{x^3+y^4}$ when $(x, y)\neq (0, 0)$ and $f(x, y)=(0, 0)$ when $(x, y)=(0, 0)$. Is this function continuous at $(0, 0)$, please? I think it is. I have tried to let $y=kx$ and $y=x^a$. How can I prove it in general, please? I think I should show that $\lim f(x, y) = 0$. But how to do it in general, please?

There are most likely simpler counterexamples, but this is easy to compute.
Even if we ignore that $f$ is not completely defined in a neighbourhood of the origin, the function $f$ is still not continuous.
$f(-\sqrt[3]{t^4},t+t^2) = \frac{-\sqrt[3]{t^4}(t+t^2)^3}{-t^4+(t+t^2)^4} = \frac{-\sqrt[3]{t^4} t^3 (1+t)^3}{t^5(t+2)(t^2+2t+2)} = -\frac{\sqrt[3]{t^4}}{t^5}\frac{(1+t)^3}{(t+2)(t^2+2t+2)}$.
We see that $\lim_{t \downarrow 0} f(-\sqrt[3]{t^4},t+t^2) = - \infty$, hence $f$ is not continuous at $(0,0)$.
