Suppose that we are given open sets $A \subset \mathbb{R}^N$ and $B \subset \mathbb{R}l^M$ and a (continuously differentiable) function $f: A \times B \rightarrow \mathbb{R}^N. If $f(\cdot ; \cdot)$ satisfies the condition
The $N \times (N+M)$ matrix $Df(x;q)$ has rank $N$ whenever $f(x;q)=0$,
then the system of $N$ equations in $N$ unknows $f(\cdot; \hat{q})=0$ is regular for almost every $\hat{q} \in B$.*
*: “Almost every” means that if, for example, we choose $\hat{q}$ according to some nondegenerate multinomial normal distribution in $\mathbb{R}^M$, then with probability 1 the equation system $f(\cdot; \hat{q})=0$ is regular. This is the concept of “genericity” in this context.
See Microeconomic Theorey (MWG) page 943.
