Lets have look at the usual polar coordinates in two dimensions, given by the mapping $$\Phi:(0, \infty)\times(-\pi,\pi)\rightarrow \mathbb{R}^2\backslash (-\infty,0]\times\{0\}, \Phi (r, \theta) = (r \sin\theta, r\cos\theta) $$
Almost everywhere are polar coordinates like this claimed to be diffeomorphisms. For example in integration, or as parametrisations of manifolds. But actually, they aren't because the inverse map is not continuous (around every point on $(-\infty,0]\times\{0\}$ in our example).
Well you could possibly fix that by restriction and using a second map but rotated, but nobody does that right? Or am I wrong? For example if you integrate over a manifold like a ball in 3D you would need to use a partition of 1 and at least 2 polar coordinate mappings to make everything correct. Why does everybody ignore that fact?
The map $\Phi$ is a diffeomorphism from $(0, \infty) \times (-\pi, \pi)$ to $\mathbb{R}^2 \setminus (-\infty, 0] \times \{0\}$. Since points in $(-\infty, 0] \times \{0\}$ are excluded, we do not care if inverse function is continuous there. In fact, the inverse function would not be defined on $(-\infty, 0] \times \{0\}$.