reg} \cap \{ \rho < 2\,C\}$ having finitely many or at most countably many critical points[;]{} moreover, different critical points of $\mu$ are located in different level sets $\{ \mu = c \}$.*]{}

Of course, if they are infinite in number, critical values can only accumulate at $- \infty$. Similarly, we may perturb $\rho$ very slightly near $\{ \rho = C\}$ so that:

-   the $\mathcal{ C}^\infty$ hypersurface $\{ \rho = C \} \cap X_{\rm
    reg}$ does not contain any critical point of $\mu$.

Finally, again thanks to Morse transversality theory, we may perturb the connected $\mathcal{ C}^\infty$ hypersurface $M \subset \partial
D$ of Lemma 3.2 in an arbitrarily small way so that[^7]:

-   [*$M$ does not contain critical points of $\mu$, and $\mu \big
    \vert_M$ is a Morse function on $M$ having finitely many or at most countably many critical points[;]{} moreover, any two different critical points of $\mu$ or of $\mu \big \vert_M$ have different critical values.*]{}

We draw a diagram, where $X_{\rm sing}$ is symbolically represented as a continuous broken line having spikes, with a level-set $\{ \mu =
\widehat{ c}\}$ which is critical for $\mu \big \vert_M$ and a single critical point $\widehat{ p} \in M 