Let $U=(0,\infty)\times (0,2\pi]$
Define a function $F:U \rightarrow \mathbb R^2$ by $$F(r,\theta)= (r\cos(\theta),r\sin(\theta)).$$
Here all $4$ partial derivatives of component functions exist and are continuous. So $F$ is continuously differentiable with non zero Jacobian $r$ at $(r,\theta)$. So by inverse function theorem, $F$ is locally invertible at every point with continuously differentiable inverse. So in particular on the positive part of real axis. But this is not true since inverse function is not continuous on the positive part of real axis.
So my Question is : what am I missing here ?