Let $f:\mathbb{C}\rightarrow\mathbb{C}$ be a continuous function bijective on an open and bounded set $\Omega$. If $f$ maps $\Omega$ to itself, that is $f(\Omega)\subseteq \Omega$ then is it necessarily true that $f$ maps the boundary of $\Omega$ to itself? The condition that $f$ is bijective is clearly necessary, since we can take $f$ to be a constant function defined on $\Omega$ otherwise. Given that $f$ is bijective and continuous, does it necessarily have to map the boundary to itself? If not, are there sufficient smoothness conditions (holomorphic?) we can impose to make it true? I realize that we may need the boundary of $\Omega$ to be smooth also, feel free to impose any kind of restrictions to make the question well defined.
Edit: By bijection on $\Omega$, I mean that the function satifies
$$f(a)\in\Omega \iff a\in\Omega$$
Hopefully, this will help to rule out some trivial counterexamples.
Best Answer
Yes. If $f$ is continuous on $\overline{\Omega}$ and $f\colon \Omega\to\Omega$ is a bijection, then $f(\partial\Omega)\subset \partial \Omega$.
Indeed, $f(\partial\Omega)\subset \overline{\Omega}$ by continuity. Suppose $f(x)=y\in \Omega $ for some $x\in\partial \Omega$. Pick a sequence $(x_n)\subset \Omega$ such that $x_n\to x$. By the bijectivity assumption $y=f(x')$ for some $x'\in \Omega$. By bijectivity & continuity, there exists a neighborhood $U$ of $y$ such that $f^{-1}(U)\cap \Omega$ is compactly contained in $\Omega$. Yet $f(x_n)\in U$ for all large $n$, a contradiction.