Given continuous (say $\mathcal C^2$) function
$f: \mathbb R^{n}\rightarrow \mathbb R$ what could be continuity properties for an isosurface of such a function (of each of its disjoint components)?
Can we expect it to be same ($\mathcal C^2$) continuous?
It seems easy to prove for $\mathcal C^0$, but I afraid to miss some nuances - multidimensionality often gives surprises.
Thank you very much for the clarification.