The implicit function theorem gives a sufficient criterium to determine if a given set is locally the graph of a function.
Let's say $f: \mathbb{R}^3 \to \mathbb{R}$ is smooth and $\partial_z f(x,y,z)$ is in the interval $[\frac{1}{2}, \frac{3}{2}]$ for every $(x,y,z) \in \mathbb{R}$.
With the implicit function theorem I can locally find functions $z:U \to R$, such that the set $M = \{(x,y,z) \ : \ f(x,y,z) = 0\}$ locally is the graph of the functions $z$.
Can I somehow patch these functions together to obtain a globally defined function $z: \mathbb{R}^2 \to \mathbb{R}$, such that $M$ is the graph of $z$?