Is it always true that $\partial f(U)=f(\partial U)$ when $f$ is holomorphic? Let $D\subseteq\Bbb C$, $f:D\to\Bbb C$ holomorphic on $D$.
Let $U$ be an open subset strictly contained in $D$: in this way $\partial U$ would be contained in $D$.
So I was asking myself if  $\partial f(U)=f(\partial U)$ is always true. 
It seems obvious simply passing to the limit; am I right?
 A: No, $\partial f(U) = f(\partial U)$ need not be true.
As an example, take $D = \Bbb C$,
$$U = \{ z : |z| < 1 \text{ and } \operatorname{Re} z > 0 \} 
$$
and $f(z) =  z^4$. Then 
$$
 f(U) = \{ z : |z| < 1 \} \setminus \{ 0 \}
$$
and 
$$
 f(\partial U) = \{ z : |z| = 1 \} \cup [0, 1)
$$
is a strict superset of
 $$\partial f(U) = \{ z : |z| = 1 \}  \cup \{ 0 \} \, .
$$

If $U$ is bounded then $\partial f(U) \subset f(\partial U)$ holds.
Proof:
$\overline U$ is compact, therefore $f(\overline U)$ is compact as
well. It follows that
$$
 \overline{f(U)} \subset f(\overline U) \, .
$$
 A (non-constant) holomorphic function is an open mapping,
therefore $f(U)$ is an open set, so that
$$
 \partial f(U) = \overline{f(U)} \setminus f(U) 
 \subset f(\overline U) \setminus f(U)  \subset f(\overline U
 \setminus U) = f(\partial U) \, .
$$

If, in addition, $f$ is injective (or more generally,
a proper map from
$U$ to $f(U)$) then equality holds.
Proof: For a proper map it holds that
$$ \tag{*}
 f(U) \cap f(\partial U) = \emptyset
$$
and therefore
$$
 f(\partial U) \subset f(\overline U) \subset \overline{f(U)} \\
 \Longrightarrow  f(\partial U) \subset  \overline{f(U)} \setminus f(U) = \partial f(U) \, .
$$
Proof of $(*)$: Assume that $w_0 \in f(U) \cap f(\partial U)$.
Let $K$ be a compact disk such that $w_0 \in K \subset f(U)$.
From $w \in f(\partial U)$ it follows that there is a sequence $(z_n)$,
$z_n \in U$, such that $z_n \to z_0 \in \partial U$ and $f(z_0) = w_0$.
Then $f(z_n) \to f(z_0) = w_0$ and therefore $f(z_n) \in K$ for
$n \ge n_0$, so that $z_n \in f^{-1}(K)$ for $n \ge n_0$.
If $f$ is a proper map then $ f^{-1}(K)$ is compact in $U$ which 
is a contradiction to $z_n \to z_0 \in \partial U$.
