If the boundary of an open connected set $\Omega$ of class $C^1$ is bounded $\Rightarrow$ $\Omega$ is bounded?

Question

Let $\Omega \subset \mathbb{R}^N$ be an open set.

I know bounded boundary doesn't imply bounded set, but what if we consider the boundary of an open connected set of class $C^1$ (i.e. the boundary $\partial \Omega$ is locally the graph of a $C^1$ function)-

If the boundary of a connected open set $\Omega$ of class $C^1$ is bounded then $\Omega$ is bounded?

Answer 1

Consider $\Omega = \{ x \in \Bbb R^n | \| x \| > 1 \}$

It is open connected, and $\partial \Omega$ is $C^1$ but $\Omega$ is not bounded