I was stuck on the following problem:
Let $f$ be analytic on $D = \{z \in \Bbb{C} : |z| < 1\}$ and $f(0) = 0$.
Define $g(z) = \begin{cases} \displaystyle \frac{f(z)}{z} & z \neq 0, \\ f'(0) & z = 0. \end{cases}$
Then which of the following option(s) is/are correct?
$g$ is discontinuous at $z = 0$ for all $f$.
$g$ is continuous, but not analytic, at $z = 0$ for all $f$.
$g$ is analytic at $z = 0$ for all $f$.
$g$ is analytic at $z = 0$ only if $f'(0) = 0$.
Can someone point me in the right direction with some explanation? Thanks in advance for your time.
EDIT: I have posted an answer .Feel free to comment if I missed anything in my answer.