Problem in understanding proof of Implicit Function Theorem from Calculus of Manifolds

Question

From the proof of the Implicit Function Theorem given in Spivak's "Calculus of manifolds" page 42, for $f: R^n \times R^m \to R^m$ which is continuously differentiable we define a $F:R^n \times R^m \to R^n\times R^m$ such that $F(x,y)=(x,f(x,y))$ then if $F^\prime(a,b)\neq 0$ for $(a,b)$ contained in $W \subset A\times B$ with $A \subset R^n , B \subset R^m $ then $F:A \times B \to W$ has a differential inverse $h:W\to A\times B$ which is of the form $h(x,y)=(x,k(x,y))$ for some differentiable function $k$ (since $F$ is of this form). What I cannot understand is that why does the function $k$ become differentiable since we haven't proved it.

Answer 1

It is part of the (multivariate) inverse function theorem that the inverse function of an $F: \>{\mathbb R}^d\to{\mathbb R}^d$ is again differentiable. This part does not come for free; some extra work is needed. This fact immediately implies that your $k$, being the second component of $F^{-1}$, is differentiable.