Real boxes and interval Jacobian enclosures

For an integer $n\ge 1$, a real box is a nonempty product $X={\prod }_{j=1}^n[a_j,b_j]\subset {\mathbb{R}}^n$ of closed real intervals with finite dyadic endpoints $a_j\le b_j$, using the interval convention of Real intervals, complex rectangles and ball enclosures. Its interior is $\mathrm{int}X={\prod }_{j=1}^n(a_j,b_j)$; it is empty if any side is degenerate.

Let $U\subset {\mathbb{R}}^n$ be open and convex, let $X\subset U$ be a real box, and let $F:U\to {\mathbb{R}}^n$ be $C^1$ in the sense of Continuously differentiable maps, local inverses, and local diffeomorphisms. An interval Jacobian enclosure on $X$ is an $n\times n$ matrix $J(X)=(J_{ij})$ of finite real intervals such that ${\mathrm{\partial }}_j{F}_i(X):=\{{\mathrm{\partial }}_j{F}_i(x):x\in X\}\subseteq J_{ij}$ for every $i,j$, with partial derivatives and Jacobian entries as in The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case. Write $\mathcal{J}(X)=\{A\in {\mathbb{R}}^{n\times n}:A_{ij}\in J_{ij}\}$ for the set of point matrices represented by this interval matrix.