I've proven problem $2-36$ and now want to assume that $g$, from problem $2-37$, is injective so that it can satisfy the conditions that $f$ does in $2-36$. I need $g$ to be continuously differentiable and to have det $g'(x,y)\neq 0$. I have two questions
- Since I have that the $2$ component functions of $g$ are continuously differentiable, does that imply that $g$ is continuously differentiable?
- Does $D_1f(x,y)\neq 0$ imply that det $g'(x,y)\neq 0$?
Thanks!
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Remember that a function $f$ is continuously differentiable in $U$ open if and only if $f$ is differentiable in $U$ and its partial derivatives are continuous in $U$. Moreover, a function $f$ is differentiable if only if all its component functions are differentiable. So...
Because $$ det\ g'(x,y)=det \begin{pmatrix} D_{ 1 }f(x,y) & D_{ 2 }f(x,y) \\ 0 & 1 \end{pmatrix} = D_1f(x,y).$$