$f$ be analytic on $D=\{z \in \mathbb C: |z|<1\}$ and $f(0)=0.$ Define $g(z)=f(z)/z;z \neq 0$ and $g(z)=f'(0);z=0.$ I was thinking about the following problem:
Let $f$ be analytic on $D=\{z \in \mathbb C: |z|<1\}$ and $f(0)=0.$ Define $g(z)=f(z)/z;z \neq 0$ and $g(z)=f'(0);z=0.$ Then which of the following options are true:  

(a)$ g$ is discontinuous at $z=0$
  (b)$g$ is continuous but not analytic at $z=0$ for all $f$
  (c)$g$ is  analytic at $z=0$ for all $f$
  (d)$g$ is  analytic at $z=0$ only if $f'(0)=0$.

My attempts: Clearly $g$ is continuous at $z=0$, since we see that $\lim_{z\to\ 0}g(z) =\lim_{z\to\ 0} f(z)/z=\lim_{z\to\ 0}f'(z)$[by L'hospital's rule as $0/0$ form]=$f'(0)=g(0).$ But here i know that i have to use Schwarz lemma but i could not use it effectively enough to make other conclusion about the other options.
Also i see that
$f$ is analytic in some neighbourhood of $0$, so i can write it as a power series in $z.$ that is $f(z)=a_0+a_1(z)+a_2(z^2)+..... $and so $g(z)=f(z)/z=a_0/z+a_1+a_2(z)+...$but now i lost the track, as see if i take $f'(0)=g(0)=0$ ,then $g'(0)={g(z)-g(0)}/(z-0)=a_0/z^2+a_1/z+a_2+...$ which clearly does not exist at $z=0$ even if we take $f'(0)=0.$ Am i going in the right direction?
I need help about options $(b),(c),(d).$Thanks in advance for your time.

I suggest using Riemann's Removable Singularity Theorem. This will make the problem rather simple
