Let's say $\mathbf{x} \in \mathbb{R}^n$ are a vector in $n$-dimension real coordinate. Another vector, $\mathbf{y} \in \mathbb{R}^n$, is a result of unknown transformation from $\mathbf{x}$, i.e. $\mathbf{y}=\mathbf{f}(\mathbf{x})$. If we only know the Jacobian matrix, i.e. $\frac{\partial \mathbf{f}}{\partial \mathbf{x}}=\mathbf{A}(\mathbf{x})$ at any $\mathbf{x}$, and the matrix $\mathbf{A}(\mathbf{x})$ is invertible and continuous everywhere, is it possible to show the existence of the mapping $\mathbf{f}$ and it's bijectivity?
It is easy to do this if $n=1$. In this case, invertibility of $A(x)$ means that it is always positive or always negative. Therefore, the mapping is simply $f(x)=f(x_0)+\int_{x_0}^{x}A(x')dx'$, which is strictly monotonous, therefore it exists and is bijective for any arbitrary invertible and continuous $A(x)$. However, I'm not sure if this is the case for $n > 1$. (I don't need to know the exact form, I just need to know the existence and the bijectivity of $\mathbf{f}$)