I am recently studying about stereographic projection and it is stated that the punctured 2-sphere $S^2\backslash \left\{0,0,1\right\}$ is diffeomorphic to $\mathbb{R}^2$. I want to prove this statement but I am stuck at proving the mapping is differentiable. May I have some helps? Here is my work:
From the construction of stereographic projection, it is clear that $f:S^2\backslash \left\{0,0,1\right\}\rightarrow \mathbb{R}^2$ defined by $$f(x,y,z)=\left(\dfrac{x}{1-z},\dfrac{y}{1-z},0\right)$$ is bijective.
Then, I cannot proceed to prove that $f$ and $f^{-1}$ are differentiable. How can I prove it?