Suppose $f:\mathbb{R}^m\times\mathbb{R}^n\rightarrow\mathbb{R}^p$ is infinitely differentiable, where $p\ge m+n$.
Define $g_{x,y}:\mathbb{R}\rightarrow\mathbb{R}^p$ by $g_{x,y}(\sigma)=f(\sigma x,\sigma^2 y)$ for all $x\in\mathbb{R}^m$, $y\in\mathbb{R}^n$ and $\sigma\in\mathbb{R}$, and suppose that:
$$g_{x,y}(\sigma)=g_{0}+g_{x,1}\sigma+g_{x,y,2}\sigma^2+g_{x,y,3}\sigma^3+O(\sigma^4)$$
as $\sigma\rightarrow 0$.
Finally define $h_{x,y}:\mathbb{R}\rightarrow\mathbb{R}^p$ for all $x\in\mathbb{R}^m$, $y\in\mathbb{R}^n$ and $\sigma\in\mathbb{R}$, by:
$$h_{x,y}(\sigma)=g_{0}+g_{x,1}\sigma+g_{x,y,2}\sigma^2+g_{x,y,3}\sigma^3,$$
so $h_{x,y}$ is a Taylor approximation to $g_{x,y}$, and a perturbation type approximation to $f$.
Now, suppose that for all $x_1,x_2\in\mathbb{R}^m$ and $y_1,y_2\in\mathbb{R}^n$:
$$\left( h_{x_1,y_1}(1)=h_{x_2,y_2}(1)\right)\rightarrow\left(x_1=x_2\wedge y_1=y_2 \right).$$
Given this, can one prove that there exists some open sets $U\subseteq\mathbb{R}^m$ and $V\subseteq\mathbb{R}^n$ with $0\in U$ and $0\in V$ such that for all $x_1,x_2\in U$ and $y_1,y_2\in V$:
$$\left( f(x_1,y_1)=f(x_2,y_2)\right)\rightarrow\left(x_1=x_2\wedge y_1=y_2 \right)?$$
Can one say anything more about $f$ than this? I.e. might it be the case that for all $x_1\in\mathbb{R}^m$ and $y_1\in\mathbb{R}^n$, there are finitely (countably?) many $x_2\in\mathbb{R}^m$ and $y_2\in\mathbb{R}^n$ such that $f(x_1,y_1)=f(x_2,y_2)$?
One potential avenue might be considering homotopies between $g_{x,y}$ and $h_{x,y}$, so I am tagging with homotopy theory.