Consider a function $\mathbf f : \mathbb R^n \to \mathbb R$ as an element of $\mathcal C^k(\mathbb R^n)$ and a fixed real number $C \in \mathbb R$. Now define the multidimensional level set $\mathcal A_C = \displaystyle\{\mathbf x \in \mathbb R^n : \mathbf f(\mathbf x) = C \}.$
Does the constrains on $\mathbf f$ imply anything for the set $\mathcal A_C$?
As an example, consider the simple case of $\begin{cases} k=0 \\ n=2 \end{cases}$ and the question about the nature of the multidimensional level set $\mathcal A_C$ as defined above.
This means that $\mathbb f$ maps the plane continuously to the real line. Also assume that there's no ball $B(\varepsilon, \mathbf x_0) = \displaystyle \{ \mathbf x \in \mathbb R^n : |\mathbf x - \mathbf x_0| < \varepsilon \}$ such that $\mathbf f$ is constant on $B(\varepsilon, \mathbf x_0)$. It seems likely that $\mathcal A_C$ consists of a countable union – which I stress could be empty – of continuous curves $\gamma \in \mathcal C(\mathbb R)$, although a such conjecture obviously have to be proven.
In the more general form, where $k, n \in \mathbb N$ is chosen arbitrary, can we say something about the nature/topology of the set $\mathcal A_C$? Does less restrictive requirements on the function $\mathbf f$ enable us to say more, for example if we only required $\mathbf f$ to be locally Lipschitz?
(In the more general form, where $\mathbf f \in \mathcal C^k(\mathbb R^n)$, it seems like a probable guess that $\mathcal A_C$ has to consist of an countable union of sets which are path-connected by $C^k$-curves. Also note that points are, in this sense, curves.)