Say you are given a one-to-one mapping $f$ from the real plane to the real plane, where $$f(x,y)=(f_1 (x,y), f_2 (x,y))$$
How do you check if f is a diffeomorphism?
Furthermore how do you restrict the domain so that it becomes a diffeomorphism if it wasn't already one?
I know $f$ is a diffeomorphism if $f$ and its inverse is continuous and differentiable.
Suppose $f$ is continuous everywhere except points $P_1,P_2,...,P_n$. If the Jacobian determinant of $f$ is non-zero for all points excluding $P_1,P_2,...P_n$ then $f$ is a homeomorphism on this new domain?
To show differentiability you need to check the continuity of the first partial derivatives of $f_1$ and $f_2$ with respect to $x,y$?
Finally, say $f$ is continuous on the real plane except the points $P_1,P_2,...,P_n$ and suppose:
$$\text{ the first partial derivatives of } f_1 \text{ and} f_2 \text{ are only continuous when } $$ $$(x,y) \text{ is not the points } P_1,P_2,...,P_n,A_1,A_2,...,A_k$$
Then $f$ is differentaible on the original domain take away these points?
Thanks in advance for your response.