Continuity via polar coordinates

Polar coordinates. For $(x,y) = (r\cos \theta, r \sin \theta)$,
$$|f(x,y)| =r|\cos^3\theta+\sin^3\theta| \leq 2r = 2 \sqrt{x^2 + y^2} \rightarrow 0$$
as $(x,y) \rightarrow (0,0)$.

Conclusion. $\lim_{(x,y)\to(0,0)}f=0=f(0,0)$: $f$ is continuous at the origin.