  In mathematical modeling, it is common to have an equation $F(p)=c$ where the
exact form of $F$ is not known.
  This article shows that there are large classes of $F$ where almost all $F$
share the same properties. The classes we investigate are vector spaces
$\mathcal{F}$ of $C^1$ functions $F:\mathbb{R}^N \to \mathbb{R}^M$ that satisfy
the following condition: $\mathcal{F}$ has ``almost constant rank'' (ACR) if
there is a constant integer $\rho(\mathcal{F}) \geq 0$ such that
rank$(DF(p))=\rho(\mathcal{F})$ for ``almost every'' $F\in \mathcal{F}$ and
almost every $p\in\mathbb{R}^N$. If the vector space $\mathcal{F}$ is
finite-dimensional, then ``almost every'' is with respect to Lebesgue measure
on $\mathcal{F}$, and otherwise, it means almost every in the sense of
prevalence, as described herein. Most function spaces commonly used for
modeling purposes are ACR. In particular, we show that if all of the functions
in $\mathcal{F}$ are linear or polynomial or real analytic, or if $\mathcal{F}$
is the set of all functions in a ``structured system'', then $\mathcal{F}$ is
ACR. For each $F$ and $p$, the solution set of $p \in \mathbb{R}^N$ is
  SolSet$(p):= \{x: F(x)=F(p)\}.$ A solution set of $F(p)=c$ is called robust
if it persists despite small changes in $F$ and $c$. The following two global
results are proved for almost every $F$ in an ACR vector space $\mathcal{F}$:
(1) Either the solution set SolSet$(p)$ is robust for almost every $p\in
\mathbb{R}^N$, or none of the solution sets are robust. (2) The solution set
SolSet$(p)$ is a $C^\infty$-manifold of dimension $d = N-\rho(\mathcal{F})$. In
particular, $d$ is the same for almost every $F \in \mathcal{F}$.
