Let $g:\mathbb{R}^n \times \mathbb{R} \rightarrow \mathbb{R}^k$ be a function whose coordinates $g_i$ are homogeneous polynomials and let $u \in \mathbb{R}^k$ and $c \in [0,1]$ and let $J_x(c)$ be the Jacobian matrix of $g(\cdot,c)$ at $x$.
Let $M(c)=\{x \in \mathbb{R}^n | g_i(x,c)=u_i, i=1\cdots,k\}$ be a space such that $J_x(c)$ is full rank at the interior of $M(c)$ for all $c \in [0,1]$. Thus $M(c)$ is a collection of spaces whose interior are smooth manifolds.
I try to investigate if the number of connected components of $M(c)$ is equal to $M(c')$ where $c' = c + \delta c$ and $\delta c$ is sufficient small.
Some of you have an idea or which tools to use in order to investigate the above property?
Thanks in advance.t