Let $B = [a -\varepsilon, a + \varepsilon] \times [b -\varepsilon, b + \varepsilon]$ be a box in $\mathbb{R}^2$. Let $\Psi_1(w_1, w_2, \mathbf{v})$ and $\Psi_2(w_1, w_2, \mathbf{v})$ be two polynomials in $n$ variables. Let us denote $\Psi = (\Psi_1, \Psi_2)$. Suppose for $\mathbf{v} \in U$, where $U$ is an open set in $\mathbb{R}^{n-2}$, we have that that $\Psi(B, \mathbf{v})$ is diffeomorphic to $B$.
What I am wodering is, is the function $$ H(\mathbf{y}, \mathbf{v}) = d( \mathbf{y}, \partial \Psi(B, \mathbf{v})) $$ a continuous function from $\mathbb{R}^2 \times U$ to $\mathbb{R}$?
Here $\partial \Psi(B, \mathbf{v})$ is the boundary of the set $\Psi(B, \mathbf{v})$, and given a set $X \subseteq \mathbb{R}^2$, $$ d(\mathbf{y}, X) = \inf_{\mathbf{x} \in X} \| \mathbf{y} - \mathbf{x} \|_{\infty}. $$
Maybe I am just missing something simple but I am having difficulty with this. Any comments or suggestions are appreciated. Thank you very much.