Finding a function with arbitrary Jacobian determinant everywhere

Question

If we have a function $g: \mathbb{R}^n \rightarrow \mathbb{R}$ and $ \forall x, g(x) > 0$, can we always find a function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ s.t. $\forall x, |\det \frac{\partial f(x)}{\partial x}| = g(x)$ under reasonable continuity / differentiability assumptions about $g$? I think it should be true, but I'm not sure how to prove it.

Answer 1

It will be enough to take $f(x_1,\ldots,x_n)=\bigl(G(x_1,\ldots,x_n),x_2,\ldots,x_n\bigr)$, where $\frac{\partial G}{\partial x_1}=g$.

Answer 2

If $g$ is continuous then it has an antiderivative $G$ you can simply take

$$ f(x_1, x_2,\dots,x_n) = (G(x_1),x_2,\dots,x_n). $$

Probably continuity of $g$ is about as general as you'd want to ask this question for.