# Multiple choice Question on Complex Analysis

Let $f$ be an analytic function defined on the unit disc in $\mathbb C$ .Then $f$ is constant if

$1.f \left(\dfrac{1}{n} \right)=0$

$2.f(z)=0$ for all $|z|=\dfrac{1}{2}$

$3.f \left(\dfrac{1}{n^2} \right)=0$

$4.f(z)=0$ for all $z\in (-1,1)$

My try:

Since in 1 and 3 ;$\dfrac{1}{n}$ and $\dfrac{1}{n^2}$ has limit point 0 which lies in the unit disc so $f(z)\equiv 0$ and hence constant and in $4$ $f$ is zero on a connected open set and hence zero identically .

So 1,3,4 are the correct options

(2) is also right, because $\{z:|z|=1/2\}$ has a limit point inside the unit disc.