Show that the function is continuous which points.
$$\displaystyle f(z) = \begin{cases} \frac{x^2}{x^2+y^2}+2i, & \text{if $z \neq 0$ } \\[2ex] 2i, & \text{if $z=0$ } \end{cases}$$
Consider $f(0)=2i$ and $\lim_{z\to 0}f(z)$ That is, we consider $z$ approach $0$ along the real axis
I have $\lim_{z\to 0}f(z)=1+2i$
Thus $\lim_{z\to 0}f(z) \neq f(0)$
Therefore $f(z)$ continuous every point on $\mathbb C$ expect point $(0,0)$
Please check my solution. If it wrong please tell me the right answer. Thank you.
Consider $f (i/n) $ to see that $f $ does not have a limit for $z \to 0$.