This is a continuation of a previous question .
Define that elements of $x=(x_1,x_2,...,x_n)$ are distinct if for each $i≠j, x_i≠x_j.$ consider a system of linear equation $Ax=b$. We want to understand the property of $(A,b)$ such that the solution $x$ given $A$ and $b$ is distinct in the sense defined above.
In a previous question, Theo kindly showed that the set of such $(A,b)$ is open and dense. Now we want to extend the result to a system of nonlinear equations
Let $f_1,...,f_n$ be some functions and $b_1,...,b_n$ be scalars. Consider a system of nonlinear equations $f_1(x_1)=b_1,...,f_n(x_n)=b_n.$ Under what condition on $f$, can we say that the set of $b$ such that the resulting $x$ is distinct is open and dense?
Can we extend Theo's argument if I can apply the inverse function theorem to this system of nonlinear equations?