Existence of a holomorphic function with specific caracteristics

Question

Is it possible to find a holomorphic function $f : D \rightarrow \mathbb{C}$ where $D$ is the $\mathbb{C}$ open unit disk such that:

1. $f$ is continuous in $\overline{D}$
2. $f (\partial D)\subset \partial D$
3. The winding number of $\partial D$ around $0$ is equal to $1$
4. $f$ is not one-to-one on $\partial D$

Answer 1

With Misha's correction, the answer is no. The maximum principle imply that $f(\mathbb{D}) \subseteq \mathbb{D}$ (note that $f$ is not constant because of condition 3). Conditions 1 and 2 imply that $f:\mathbb{D} \to \mathbb{D}$ is proper, so it has degree $n$ for some $n$. But condition 3 implies that $n=1$.

EDIT : In other words : by the Schwarz reflection principle, $f$ extends analytically on a neighborhood of $\overline{\mathbb{D}}$. Analyticity forces $f$ to be monotonically increasing on $\partial \mathbb{D}$, and condition 3 then implies that $f$ is one-to-one on $\partial \mathbb{D}$.