Proving implicit function theorem with the fixed point theorem I am trying to solve an exercise in Kolmogorov's analysis text asking for a proof of the implicit theorem using the fixed point theorem. I am struggling with how to get started on this, though. It's clear that I need to define a mapping, $T$, demonstrate that $\rho(Tx, Ty) = \alpha \rho(x,y)$ for some $\alpha < 1$, and then conclude that because $T$ is a contraction mapping, it has a fixed point, $x$, such that $Tx = x$. I know of only one proof of the implicit function theorem, though, and it in no way uses any of these facts. (I proved it at a time when I didn't know what a contraction mapping was.) 
Any help on this would be greatly appreciated. 

Hint:
For express $x_0$ over other variables, define $g(X) = \frac{f(X)}{\max_{[x_0-\epsilon,x_0+\epsilon]} f'}$ can be good candidate, since $\max f'$ is finite due to theorem hypothesis.
