Consider the relation $f(x,y)=0$, with $f:\mathbb{R}^n\times\mathbb{R}^m\rightarrow \mathbb{R}^n$. The (standard) implicit value theorem gives you conditions for the existence of a function $g:B\rightarrow \mathbb{R}^m$ such that $f(x,g(x))=0$ for all $x$ in some open ball $B$ around a given point $a\in\mathbb{R}^n$. In addition, if $g$ exists then it inherits certain smoothness properties from $f$, i.e. if $f\in\mathcal{C}^k$ then $g\in\mathcal{C}^k$.
Suppose you already know there that there exists a $g$ such that $f(x,g(x))=0$ for all $x$ in a given open set A. What properties smoothness properties can $g$ be expected to have? More specifically if $f\in\mathcal{C}^k$ is $g\in\mathcal{C}^k$ and if $f$ is globally Lipschitz continuous is $g$ globally Lipschitz continuous?
Thanks in advance.