Let $f_i:\mathbb{R}^{n+1}\to \mathbb{R}^n$ for $i\in \mathbb{N}$ be collection of smooth maps such that,
- $f_i(0)=0$ for all $i$;
- The Jacobian $Df_i$ is invertible at $0$ for all $i$;
- There exists $M>0$ such that $|f_i|^2\leq M$ for all $i$ (i.e. $M$ is a uniform (independent of $i$) bound on the size of all $f_i$).
By (1) and (2), we can apply the implicit function theorem to all the $f_i$ at $0$: as a consequence, there exist neighborhoods $U_i\subset \mathbb{R}^n$ of $0\in \mathbb{R}^n$ and smooth maps $g_i:U_i\to \mathbb{R}$ such that $g_i(0)=0$ and $f_i(x,g_i(x))=0$ for all $x\in U_i$.
Let $B_{r_i}(0)\subset g_i(U_i)$ be the largest ball, centered at $0\in \mathbb{R}$, which is contained in the image $g_i(U_i)$. Note that the implicit function theorem ensures that $r_i>0$ for each $i$.
Question 1: Does there exist $R>0$ such that $r_i\geq R$ for all $i$? In other words, can the size of $g_i(U_i)$ go to zero as $i\to \infty$?
Question 2: More generally, what controls the size of the neighborhoods $U_i$ (or the size of the images $g_i(U_i)$) whose existence is given by the implicit function theorem?