TITLE: Graphical representation of a complex function
QUESTION [0 upvotes]: For the given function:
$$
f(z)=\left\{\begin{matrix}
1 &  \text{if}\ z = 0\\ 
 \frac{{}\bar{{z}}}{z} & \text{if}\ z\neq 0 
\end{matrix}\right.
$$
How would I prove that $ f(z) $ is continuous for all of at the point where $z=0$?
My thoughts:
You would have to show that the limit as $z$ approaches 0 is equal to 1. How would I go about doing this? Would the Cauchy-Riemman Equations be the best approach for this type of question?

REPLY [1 votes]: Cauchy-Riemann equations provide us with a necessary condition for the differentiability of a function. Whereas to check the continuity of a function (say) $f$ at some point $z_0$ you just need to check the following 3 conditions:
(1) $f$ is defined at the point $z_0$
(2) $\lim _{z\to z_0} f(z)$ exists
(3) $\lim_{z\to z_0} f(z)=f(z_0)$
Now comming to your question, condition (1) is satisfied by $f$ in the problem. To compute the value of the limit you can convert the function into polar form.
i.e. Take  $z=re^{i\theta}$ and letting $r\to 0$
$$lim_{r\to 0}\frac{re^{-i\theta}}{re^{i\theta}}=e^{-2i\theta}$$
Now since the limit value is different for different choices of $\theta$, the above limit does not exist.
i.e $\lim _{z\to 0} f(z)$ does not exists
Hence, the condition (2) is not satisfied and we can thus conclude that $f$ is not continuous at $z=0$
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