Determine the points of complex plane, where the function is analytic $f(z)=z^{2}\overline{z}$.
My approach: Note that, $$\dfrac{\partial f(z)}{\partial \overline{z}}=z^{2}=0$$ this implies that $z=0$, is the only point where the function is analytic. This is correct?
$f(z) = z^2 \overline{z} = z |z|^2$. Note $z$ is entire, so the set where $f$ is analytic is solely determined by $|z|^2$. For any neighborhood of any complex number $z$, this function ($|z|^2$) has zero imaginary part. Assume $f$ is analytic in some (open) set. Now, without loss of generality, assume our small neighborhood is connected, then we may conclude that this function itself is constant in that neighborhood, which is a contradiction.