Prove that the differentiable function F(x,y,z)=\left (  f(x,y,z) ,g(x,y,z) , f(x,y,z)+g(x,y,z) \right) can never have a differentiable inverse.
My attempt : I notice that \dim dom F=3 and that \dim Im F =2. Hence F is not injective, hence F doesn't have an inverse, so it cannot have a differentiable inverse.
I'm not sure my attempt is good.
Maybe I should have done it using the Inverse Function Theorem, but if the Jacobian of F is not invertible, I can't say anything whether the existence of an inverse of F, I believe. Hmm but according to my memory of Linear Algebra, if the matrix of a function is not invertible then the function is not injective so it doesn't have an inverse.
What do you think?