Defining functions for connected sets Let $\Omega \subset \mathbb{R}^n$ an open, bounded and connected set with a $C^2$ boundary and a function $\rho \in C^2(\mathbb{R}^n)$ such that
$$ \Omega = \{ x \in \mathbb{R}^n : \rho(x) < 0 \},\quad \partial \Omega = \{ x \in \mathbb{R}^n : \rho(x) = 0 \},\quad |\nabla \rho(x)|=1\ \ \text{on}\ \partial \Omega.$$
My question is the following : is there $\epsilon > 0$ small enough such that the sets
$$  \{ x \in \mathbb{R}^n : \rho(x) <  \epsilon \},\quad \{ x \in \mathbb{R}^n : \rho(x) < -\epsilon \}$$
are still connected ?
My intuition tells me that it is true because if a new connected component appeared for any $\epsilon$, it would mean that there exists a critical point $x_0$ on the boundary for which $\nabla \rho(x_0) = 0$. However, I have troubles to prove this rigoureously. 
What do you think ?
 A: The function $f(t) = e^{-t^{2}(1 + \cos \pi t)}$ is smooth, even, and satisfies
$$
e^{-2t^{2}} \leq f(t) \leq 1\quad\text{for all real $t$,}
$$
with the lower equality achieved at every even integer and the upper achieved at every odd integer.

If $\|x\|$ denotes the Euclidean norm on $\mathbf{R}^{n}$, then
$$
\rho(x) = f\bigl(\|x\|\bigr)
$$
is smooth and positive, so the sub-level set $\Omega = \{t : \rho(t) < 0\}$ is empty, but for every $\epsilon > 0$, the sublevel set $\Omega_{\epsilon} = \{t : \rho(t) < \epsilon\}$ has infinitely many connected components.
(It's straightforward to modify this example so that $\Omega$ is non-empty; for example, take
$$
f(t) = \frac{t^{2} - c^{2}}{t^{2} + 1}e^{-t^{2}(1 + \cos \pi t)},
$$
with $c$ chosen so that $|f'(c)| = 1$.)

That said, it looks possible to prevent this type of counterexample by assuming $\rho$ has a positive lower bound outside some compact set $K$. Here's a sketch.
Since the zero set of $\rho$ is $\partial\Omega$ and $|\nabla\rho| = 1$ on $\partial\Omega$, $0$ is not a critical value of $\rho$. If $\rho$ has a positive lower bound outside $K$, there is an $\eta > 0$ such that $\rho$ has no critical values in $(-\eta, \eta)$. (Otherwise, there is a sequence of critical points $(x_{k})$ in $K$ such that $\rho(x_{k}) \to 0$; by Bolzano-Weierstrass, we may assume $(x_{k})$ converges to a point $x$ of $K$. Continuity of $\rho$ implies $\rho(x) = 0$, while continuity of $\nabla\rho$ implies $x$ is a critical point, contrary to the hypotheses in the question.)
The claim is now that if $|\epsilon| < \eta/2$, then the sublevel set $\Omega_{\epsilon}$ has the same homotopy type as $\Omega$. (In particular, if $\Omega$ is connected, then $\Omega_{\epsilon}$ is connected.) For this, borrow from Morse theory the idea of looking at the flow of $-\nabla\rho/|\nabla\rho|^{2}$ (in the complement of the critical set of $\rho$, which contains the neighborhood
$$
\Omega_{\eta}\setminus\overline{\Omega}_{-\eta}
  = \{x : |\rho(x)| < \eta\}
$$
of $\partial\Omega$). If $|t| < \eta/2$, then $|\epsilon - t| < \eta$, and the time-$t$ flow maps the level set of $\rho$ at height $\epsilon$ to the level set of height $\epsilon - t$, and continuously retracts $\Omega_{\epsilon}$ to $\Omega_{\epsilon-t}$. (Milnor's Morse Theory contains more details, if memory serves.)
