<s>[INST] Partial Differentiation Ex 2.2 Eng Mathematics by H.K. Das Test for continuity when
$f(x,y) = \frac{x^3\,y^3}{x^3+y^3}$ when $x\neq0, y\neq0$ and f(x,y)=0 when $x=0,y=0$. 
 [/INST] The domain of $f$ is $D=\mathbb{R^2}\setminus \{(x,-x):x\in\mathbb{R}\}$.
In $D$, let $y=x$ then, as $x\to 0$, also $y\to 0$ and
$$f(x,y)=\frac{x^3y^3}{x^3+y^3}=\frac{x^3}{2}\to 0.$$
In $D$, let $y=-x+x^4$ then, as $x\to 0$, also $y\to 0$ and
$$f(x,y)=\frac{x^3y^3}{x^3+y^3}=\frac{x^3(-1+x^3)^3}{1+(-1+x^3)^3}=\frac{-x^3+o(x^3)}{1-1+3x^3+o(x^3)}\to -\frac{1}{3}.$$
Since the two limits are different it follows that $f$ cannot be extended to a continuous function in $D\cup\{(0,0)\}$.
</s>