My question follows up with an additional remark from Spivak's proof of Inverse Function Theorem. The problem I have is the statement which immediately follows the If the theorem is true for $λ^{−1}∘f$ , it is clearly true for f... statement (from the link I've posted), in which Spivak assumes "at the outset" that $λ$ is the identity function, i.e. $λ=I$, while $λ$ was clearly defined as $λ=Df(a)$.
How can he even assume this without loss of generality? He's basically limiting himself to functions $f$ such that $Df(a)=I$. Did I get this wrong?
If $\mathcal J Df(a)\neq 0$ then the linear transformation $Df(a):=\lambda:\mathbb R^n\to \mathbb R^n$ is invertible in some neighborhood $U\ni a$. Note that $D\lambda(x)=\lambda$ since $\lambda$ is a linear transformation. The same is true of course, for $\lambda^{-1}.$
Now consider $g:=\lambda^{-1}\circ f.$ We have then by the chain rule,
$Dg(a)=D\lambda^{-1}(f(a))\circ Df(a)=\lambda^{-1}\circ Df(a)=I.$
If the theorem is true for $g$ then $g$ is invertible (in some neighborhood of $a$) and so $f$ is also invertible. Indeed, $g^{-1}=f^{-1}\circ\lambda\Rightarrow f^{-1}=g^{-1}\circ\lambda^{-1}.$
So we may as well assume that $Df(a)=I$ in the first place.