Consider the parametrisation of $S^2\setminus\{(0,0,1)\}$ given by $F$, where $F(x,y)=(\frac{2x}{1+r^2},\frac{2y}{1+r^2},\frac{r^2-1}{r^2+1})$ with $r^2=x^2+y^2$. The parameter set is $\mathbb R^2\setminus\{0\}$.
I want to show that $F$ has a continuous inverse by the Inverse Function Theorem. Although I am having difficulty seeing how exactly I should apply it.
We recall that the Inverse Function Theorem states that:
"If $F$ is a continuously differentiable function from an open set of $\mathbb R^n$ into $\mathbb R^n$, and the total derivative is invertible at a point $p$ (i.e. the Jacobian determinant of $F$ at $p$ is non-zero) then $F$ is invertible near $p$."
It's not hard to see that $F$ is $C^\infty$ by noting that each of its components are $C^\infty$. But in order to apply the Inverse Function Theorem $F$ would need to be a map of the form from $\mathbb R^n$ to $\mathbb R^n$ so as to be able to actually determine the Jacobian determinant. However, $F:\mathbb R^2\to\mathbb R^3$, and so we cannot form the Jacobian determinant. How, then, am I to apply to Inverse Function Theorem to see that $F^{-1}$ is also $C^\infty$?