This function is continuous at (0,0). Consider the function in polar form,put $x=rcos\theta$ and $y=rsin\theta$ in the given function, you will get $f(r,\theta) = r(cos\theta-sin\theta)(1+sin\theta.cos\theta)$. As $x \to 0$ and $y \to 0$, limits in polar coordinates becomes $r \to 0$ and no limit on $\theta$ , but as $r \to 0$, your function $f(r,\theta) = r(cos\theta-sin\theta)(1+sin\theta.cos\theta) \to 0$ whatsoever value $\theta$ takes.Therefore,$limit_{(x,y)\to(0,0)} \frac{x^3-y^3}{x^2+y^2} =0$ and the exact value at (0,0) is also $0$, hence the function is continuous at $(0,0)$.