Independence of variables in curvilinear coordinate systems

Question

Let $U$ be a connected open subset of $\Bbb{R}^n$, and let $(\xi_1,\dots,\xi_n)$ be a curvilinear smooth ($C^\infty$) coordinate system on $U$. Suppose $1\leq k<n$. A smooth function $f:U\rightarrow\Bbb{R}$ for which $$ \frac{\partial f}{\partial\xi_{k+1}},\dots,\frac{\partial f}{\partial\xi_n}\equiv 0\quad (\star) $$ does not necessarily admit a global representation of the form $g(\xi_1,\dots,\xi_k)$ on the whole $U$ even if the coordinate system $(\xi_1,\dots,\xi_n)$ coincides with the Euclidean system $(x_1,\dots,x_n)$. A typical example is $$ f(x,y)= \begin{cases} 0\quad\quad\,\,\,\, \text{if } x\leq 0\\ {\rm{e}}^{\frac{-1}{x}}\quad\,\,\,\, \text{if } x>0, y>0\\ -{\rm{e}}^{\frac{-1}{x}}\quad \text{if } x>0, y<0\\ \end{cases} $$ which is a smooth function on $U:=\Bbb{R}^2-[0,\infty)$ with $\frac{\partial f}{\partial y}\equiv 0$, but cannot be written as $f(x,y)=g(x)$ for any $g:\Bbb{R}\rightarrow\Bbb{R}$. Two observations regarding this example:

1. The function $f$ above is not analytic ($C^\omega$).
2. Defining a function $g$ of a single variable by setting $g(x_0)$ to be the value of $f$ on the vertical line $x=x_0$ does not work because there are vertical lines in $\Bbb{R}^2$ whose intersections with the domain $U$ of $f$ is disconnected, and thus $f$ may attain more than one value on $\{x=x_0\}\cap U$.

Going back to the original setting, it is not hard to show that given a curvilinear coordinate system $(\xi_1,\dots,\xi_n)$ on $U\subseteq\Bbb{R}^n$ and a smooth function $f:U\rightarrow\Bbb{R}$ satisfying $(\star)$, it may be written as $g(\xi_1,\dots,\xi_k)$ on the entirety of $U$ if either 1) $f$ is analytic or 2) any coordinate surface $\{\xi_{1}=a_1,\dots,\xi_k=a_k\}$ in $U$ is connected.

My Question: Are there conditions on the function $f$ or on the coordinate system that guarantee the existence of a global representation of the form $g(\xi_1,\dots,\xi_k)$? Also is there a term for curvilinear coordinate systems $(\xi_1,\dots,\xi_n)$ for which all coordinate surfaces (i.e. the level sets of functions of the form $(\xi_{i_1},\dots,\xi_{i_m})$) are connected?