Vtyjkyuk

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 $mathcal A_C = displaystylemathbf x in mathbb R^n : mathbf f(mathbf x) = C .$

Does the constrains on $mathbf f$ imply anything for the set $mathcal A_C$?

As an example, consider the simple case of $begincases k=0 \ n=2 endcases$ 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 $B(varepsilon, mathbf x_0) = displaystyle mathbf x - mathbf x_0$ 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.)