Existence of zero for sequence of functions for which the limit function has a zero Let $f_n \colon \mathbb R^d \to \mathbb R^d$ be a sequence of smooth functions and let $f \colon \mathbb R^d \to \mathbb R^d$ be its limit such that $\lim_{n \to \infty} f_n(x) = f(x)$ for all $x \in \mathbb R^d$, which is also smooth. I assume that there exists $x_0 \in \mathbb R^d$ such that $f(x_0) = 0$ and $\det J_f(x_0) \neq 0$, where $J_f$ denotes the Jacobian matrix of the function $f$.
I would like to prove (by also adding other assumptions if necessary) or disprove that there exists $N > 0$ such that for all $n > N$ there exists $x_n \in \mathbb R^d$ (not necessarily unique) which satisfies $f_n(x_n) = 0$ and $\lim_{n \to \infty} x_n = x_0$.
In the one-dimensional case ($d = 1$) I proved that the result holds. However, for higher dimensions $d > 1$ this seems either non-trivial or false.
 A: Here, we construct a counter-example for $d = 2$.
For each $n$, choose a smooth function $\varphi_n : \mathbb{R} \to \mathbb{R}$ so that
$$ \min\{0,x\} \leq \varphi_n(x) \leq \min\{\tfrac{1}{2n},x\}. $$
In particular, $\varphi_n(x) = x$ for $x < 0$ and $ 0 \leq \varphi_n(x) \leq \frac{1}{2n}$ for $x \geq 0$. Then define $R_n$ and $g_n$ by
\begin{align*}
R_n\begin{bmatrix} x \\ y \end{bmatrix}
&= \begin{bmatrix} \cos\tfrac{2}{n} & -\sin\tfrac{2}{n} \\ \sin\tfrac{2}{n} & \cos\tfrac{2}{n} \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} \\
g_n\begin{bmatrix} x \\ y \end{bmatrix}
&= \begin{bmatrix} x \\ \varphi_n\bigl(y - n^3 x^2 + \tfrac{1}{n}\bigr) + n^3 x^2 - \tfrac{1}{n} \end{bmatrix},
\end{align*}
and then $f_n$ by
$$ f_n = R_n^{-1}\circ g_n\circ R_n. $$
Then it is possible (although a bit tedious) to prove that $f_n(x) \to x$ as $n \to \infty$ for each $x \in \mathbb{R}^2$, but nevertheless $0 \notin f_n(\mathbb{R}^2)$ for any $n$.
To help understand how this counter-example works, I included an animation showing how the image of $f_n$ behaves as $n$ grows:1)

Here, blue curves are the images of the circles $r=\text{constant}$ under $f_n$ and the orange curves are the images of the lines $\theta=\text{constant}$ under $f_n$, where $(r,\theta)$ is the polar coordinates.
1) In this animation, the frames are actually generated by considering $n$ as a real parameter.
