ABSTRACT
With all of the necessary technical results now at our disposal, wemay prove, in stages, the inverse function theorem. We begin with the statement of the theorem.
Theorem 5.7 (The Inverse Function Theorem) Let f : Dk → Rk beC1 and suppose f ′(a) is invertible for some a ∈ Dk . Then there exist open sets U ⊂ Dk and V ⊂ Rk with a ∈ U such that f maps U one-to-one and onto V . Moreover, the function g : V → U given by g = f−1 is differentiable at each y0 ∈ V with g′(y0) =
[ f ′(x0)
]−1 where x0 = g(y0). The theorem is surprisingly difficult to prove. To do so we will need a series of five lemmas, which we now establish. The first one gives a convenient means, in a different but equivalent manner to that given in Chapter 6, for characterizing when a function is continuously differentiable at a point.
