Let $f: \mathbb{R}^{3} \rightarrow \mathbb{R}$ be a smooth function and let $\Sigma=f^{-1}(0)$ (assumed not empty). Show that if the differential $D f_{p} \neq 0$ for all $p \in \Sigma$, then $\Sigma$ is a smooth surface in $\mathbb{R}^{3}$.

Is $\left\{(x, y, z) \in \mathbb{R}^{3}: x^{2}+y^{2}=\cosh \left(z^{2}\right)\right\}$ a smooth surface? Is every surface $\Sigma \subset \mathbb{R}^{3}$ of the form $f^{-1}(0)$ for some smooth $f: \mathbb{R}^{3} \rightarrow \mathbb{R}$ ? Justify your answers.