In differentiability

Question

Let $f(x,y)$ and $g(x,y)$ are differentiable functions in $x$ and $y$. Suppose $f(x,y) = F(g(x,y))$.My question, Is $F$ differentiable function?!.

Answer 1

I don't think so. Take $g:t\rightarrow t^2$ and $F:t\rightarrow |t|$. Then $f=t\rightarrow t^2$, $g$ and $f$ are differentiable at $0$, but $F$ is not.

You'll need another property, such as local invertibility of $g$ in a neighborhood of $(x,y)$, in which case it should result directly from the invert function theorem : http://en.wikipedia.org/wiki/Inverse_function_theorem