This may be a trivial question, but here goes:
Suppose a semi-explicit differential-algebraic equation (DAE) system is defined as follows: $ \begin{align} &\dot x = f(x,z,\theta),\qquad x(0) = x_{0}\\ &g(x,z,\theta) = 0 \end{align}$ where $x \in \mathbb{R}^{n}$, $z \in \mathbb{R}^{m}$ and $\theta \in \mathbb{R}^{p}$ (a constant time-invariant vector). Suppose also that $g(\cdot)$ is at least once differentiable, and the Jacobian of $g(\cdot)$ with respect to $z$, which we call $g_{z}$ is nonsingular everywhere.
The vector $z$ is dependent, in that it is uniquely determined by $x$ and $\theta$. Its value is computed by solving the second implicit equation $g(\cdot)=0$.
Now, I want to perturb the value of $z$ by some constant quantity $d_{z} \in \mathbb{R}^{m}$, as follows: $ \begin{align} &\dot x = f(x,z+d_{z},\theta)\\ &g(x,z+d_{z},\theta) = 0 \end{align}$
Question: Can I do so for any arbitrary value of $d_{z}$ without making the system of equations inconsistent?
My partial reasoning: I'm guessing I can, because the implicit function theorem tells me given a nonsingular Jacobian $g_{z}$, there is a neighborhood around some point ($\bar x$, $\bar z$, $\bar \theta$) which is the open-set denoted $X$, there exists a function $h$ such that: $ g(x,h(x,\theta),\theta) = g(\bar x,\bar z, \bar \theta), \qquad (x,\theta) \subset X $ Therefore if we take ($\bar x$, $\bar z$, $\bar \theta$) to be the solution point, we can write the explicit function: $ z = h(x,\theta), \qquad (x,\theta) \subset X $ Therefore, in the neighborhood defined by $X$, the system can be written: $ \begin{align} &\dot x = f(x,h(x,\theta),\theta)\\ &z = h(x,\theta) \end{align}$ And I have a feeling that the perturbed system should be consistent, but I don't know how show this rigorously: $ \begin{align} &\dot x = f(x,h(x,\theta) + d_{z},\theta)\\ &z = h(x,\theta) + d_{z} \end{align}$