I am new to manifolds, and would like to understand whether I have the correct proof. As stated above, I want to prove that $M:=\{(x,y,z) \in \mathbb R^{3}: z=xy\}$ is a $2-$dim $C^{1}-$Manifold
My idea:
Define:
$\varphi: \mathbb R^2 \to M$, $(x,y)\mapsto (x,y,xy)$
Note that $\mathbb R^{2}$ is open.
It is clear that $\varphi$ is bijective. Furthermore, $\varphi$ is continuously partially differentiable in its coordinates and therefore it is continuously differentiable which thus implies that $\varphi $ is continuous.
On the issue of continuity of $\varphi^{-1}$, I can simply state the identity function is continuously differentiable and therefore $\varphi^{-1}$ is continuous.
Lastly $D\varphi(x,y)=\begin{pmatrix} 1 & 0 \\ 0 & 1 \\ y & x \end{pmatrix}$ therfore, the rank is $2$ and subsequently we have a $2-$dim $C^{1}-$Manifold as we wished.
Questions:
$1.$ How can I prove continuity of $\varphi^{-1}$ without using the fact that it is continuously differentiable
$2.$ Is my map $\varphi$ considered global since I have covered the whole of $M$ with only one map. In other words, what separates a global map from a local map?