Let $f\in C^{2}(U)$ where $U\subset \mathbb{R^n}$and $\partial U$ is $C^2$ If we let a level set of $a$ : $A=\left\{ x\in U\ :\ f(x)=a\right\} $ for a fixed $a$ in image of $f $, then $A$ is a curve.
How can we make sure that $A$ is $C^2$-curve?
It looks obvious, but is difficult to make sure.