Complex Analysis: Inverse function is holomorphic

[masterslave]

Homework Statement

The problem is from Sarason, page 44, Exercise IV.14.1.

Let f be a univalent holomorphic function in the open connected set G, and let g be the inverse function.
Assume that f(G) is open, that g is continuous, and that f\prime\neq 0\forall z\in G. Prove g is holomorphic.

Homework Equations

The Attempt at a Solution

I've tried 3 different methods, failing with each. I'd appreciate any insight into the problem.
My professor recommended looking at the calculus proof of the derivative of an inverse, and the following:
Start with w_{0}\in f(G), and with h small, look at \frac{(g(w_{0}+h)-g(w_{0}))}{h}.

Because f(G) is open, we know there is a \delta>0\ni \left|h\right|<\delta gives us w_{0}+h\in f(G).
So with h that small, we have w_{0}+h=f(z) for some z \in<br /> G.

 

[foxjwill]

You're off to a good start. I'd suggest also defining z_0\equiv f^{-1}(w_0) and then using that along with the other defined variables to write out the definitions, in terms of \epsilons and \deltas, of f being differentiable, and f' and g' being continuous.