Let us consider the following bijective applications: $$f_{k}:A_{k}→B_{k}, k=1,2,\ldots,m$$ We construct the following application: $$f=(f_1,f_2,\ldots,f_m):A_1×A_2\times\cdots ×A_m→B_1×B_2\times\cdots \times B_m$$
Hence $f$ is a vector-valued function from $A_{1}×A_{2}\times\cdots ×A_{m}$ to $B_{1}×B_{2}\times \cdots ×B_{m}$. If the determinant of the Jacobian matrix is nonzero everywhere, then by the Inverse function theorem, for every point $p$ in $A_{1}×A_{2}\times\cdots×A_{m}$, there exists a neighborhood about $p$ over which $f$ is invertible. My question is: Find sufficient and necessary conditions in which $f$ is invertible over its entire image.