Let $\Omega \subset \mathbb R^n$ be open, bounded and with smooth boundary. Then $\partial \Omega \subset \mathbb R^n$ is a closed and compact $(n - 1)$-dimensional manifold. Let $\nu : \partial \Omega \longrightarrow \mathbb S^{n - 1}$ be the unique continuous unit outer normal vector field.
Question: Is $\nu$ surjective?
I tried to fiddle around with a little bit of antipodal constructions (if I can show, that the antipodal vector is in the image, I could maybe have used the mean value theorem) or with relative topologies (assume $z \in \mathbb S^{n - 1}$ is not in the image of $\nu$. Then maybe the mapping degree helps leading to a contradiction since a hole would arise in the image). However, I was not able to come to a conclusion.
I would be grateful for ideas and/or references. Thanks in advance.
Here's an approach you should think through: Choose $w\in S^{n-1}$ and consider the function $f\colon \Omega\to \Bbb R$ defined by $f(x)=x\cdot w$. By compactness of $M=\partial\Omega$, the function attains both a maximum and a minimum on $M$. Note that $\nu(p)=+w$ when $f$ has a local maximum at $p$ and $\nu(p)=-w$ when $f$ has a local minimum at $p$.