Function $F=(f_1,f_2)$. $f_1$ and $f_2$ are differentiable real functions $\mathbb R^2\to\mathbb R$. Consider a system of equations:
$$f_1(x_1,x_2)=c_1$$
$$f_2(x_1,x_2)=c_2$$
What is the sufficient and necessary condition of that $x$ has unique solution for any $c$?
One may conjecture that $(x'-x)(F(x')-F(x))>0$ or $<0$ for any $x\neq x'$.
We know that in one dimension, To ensure that $g(x)=c$ have a unique solution for any constant $c$, $g$ must be either strictly increasing or decreasing. Does this idea extend to multi-variate functions?
Best Answer
You are considering a differentiable function $F: \mathbb R^2 \rightarrow \mathbb R^2$.
You are asking if there is a unique solution of $F$ for any element in the target set $\mathbb R^2$.
Hence $F$ must be both injective and surjective.
So $F$ must be a differentiable bijective map from $\mathbb R^2 \rightarrow \mathbb R^2$. Don't complicate yourself with $f_1$ and $f_2$s.
If you think about it, isn't that exactly what monotonicity says in 1D functions?