I was reading up on the inverse function theorem and I was wondering if the following thought I had was correct:
If $f: \mathbb{R}^n \to \mathbb{R}^n$ $g: \mathbb{R}^n \to \mathbb{R}^n$ are eachothers inverse functions and $f,g$ are both continuously differentiable, does that imply that the the derivatives $f'$ and $g'$ are invertible themselves? And more specifically is $f'$ the inverse of $g'$ and vice versa?
My first reaction would be yes since it follows that the Jacobian of the composite functions is simply the identity. Therefore the Jacobians of $f$ and $g$ are inverses. Is my reasoning flawed?
By the chain rule $$ D (f \circ g) ({\bf x}) = Df(g({\bf x})) Dg({\bf x}) $$ and for the identity operator $$ DI({\bf x}) = I $$ Since $f \circ g = I$, what this tells you is that for any ${\bf x} \in \mathbb{R}^n$, the matrix $Df(g({\bf x}))$ and $Dg({\bf x})$ are inverses of each other and, by symmetry, the same is true for $Dg(f({\bf x}))$ and $Df({\bf x})$.