I'm studying calculus with the textbook "Calculus" by James Stewart. I was curious about the sign of the Jacobian when a change of variables occurs. Inside the text's boundary, there doesn't seem to be much room for the Jacobians to misbehave. (The change of variables formula restricts the Jacobian to be nonzero; since the transformation must be continuously differentiable the Jacobian can't change its sign throughout the domain.)
However, it is not that hard to find a continuously differentiable transformation that makes the Jacobian zero, like $(u,v)\mapsto(u^{3},v^{3})$. Through some further reading at Wikipedia I found that the nonzero restriction could be dropped. A natural question followed, which is the question in the title. I know that taking the absolute value of $\det\rm{J}$ will resolve any complications regarding the integration itself — I'm just curious about the behavior of Jacobians.
To restate my question (more formally),
Question: Let $U$ be an simply connected open set in $\mathbb{R}^{n}$ and $\varphi:U\rightarrow\mathbb{R}^{n}$ an injective differentiable function with continuous partial derivatives. Then, does $$(\forall\rm{u}\in U,\det(\rm{J_{\varphi}(u)})\le0)\vee(\forall\rm{u}\in U,\det(J_{\varphi}(u))\ge0)$$ hold for all $\varphi$ satisfying the conditions?
I tried building some counterexamples, but they seemed to fail due to the condition that $\varphi$ must be injective.