The continuity of $f(z)=\left\{\begin{smallmatrix}\frac1{|z|^2}(\operatorname{Re}z)^2(\operatorname{Im}z);&z\ne 0\\0;&z=0\end{smallmatrix}\right.$

Question

It is an advance complex question… Need you suggestion kindly help me to describe the continuity of this $f(z)$ at all point of $\mathbb C$.

Answer 1

Write $z=x+iy$ and so the top part of the equation becomes $f(x+iy)=\frac{x^2y}{x^2+y^2}$. Can you use this to determine the continuity of $f$ at $z=0$? i.e. when $(x,y)=(0,0)$?