Let $U$ denote the unit open disc centred at 0. Let $f : U \backslash \{0\} \longrightarrow \mathbb{C}$ be an analytic function. Assume that $\lim_{z \longrightarrow 0} z f(z) = 0$.\\
(A) $\lim_{z \longrightarrow 0} |f(z)|$ exists and is in $\mathbb{R}$.\\
(B) $f$ has a pole of order 1 at 0.\\
(C) $zf(z)$ has a zero of order 1 at 0.\\
(D) There exists an analytic function $g : U \longrightarrow \mathbb{C}$ such that $g(z) = f(z)$ for every $z \in U \backslash \{0\}$.