Suppose I have a function $\phi_a(r)$, where $r \in \mathbb{R}^n$ denotes a real n-dim. vector, and where $a \in \mathbb{R}^p$ denotes a set of additional real parameters.
Suppose that for a given $a$ I have $r_0$, solving
$\nabla_r\,\phi_a(r)|_{r = r_0} = 0$
For fixed $a$ the set of all solutions is denoted by
$E_a[\phi] = \{r: \, \nabla_r\,\phi_a(r) = 0 \}$
I start with the restriction that there are only finitely many solutions in $E_a[\phi]$.
Suppose I start with parameters $a_0$ and change the parameters by an infinitesimal amount
$a_0 \to a = a_0 + \delta a$
In most cases this will change the solutions in $E_{a_0}[\phi]$ by an infinitesimal amount, too (there are counter-examples, e.g. for $\phi_a(r) = |r|^a$ which changes its behaviour at $a = 0, 1$)
So we will have
$r_0 \to r = r_0 + \delta r$
For one solution $\nabla_r\,\phi_a(r)|_{r = r_0} = 0$ varing $a$ defines a curve $C(a)$ in $\mathbb{R}^n$
$C: \mathbb{R}^p \to \mathbb{R}^n$
My questions are:
What are the conditions for the function $\phi$ such that nothing unexpected occurs, that by varying $a$ no solution in $E[\phi]$ disappears, i.e. that a curve $C(a)$ does not end a some point in the parameter space?
What can be said locally, i.e. for infinitesimal changes?
What can be said globally, i.e. for the whole curve?
What is the mathematical discipline dealing with this problem?