Let $U\subset \mathbb{R}^n$ and let $F:U\to \mathbb{R}^n$. The 'classical' inverse function theorem gives a sufficient condition for the existence and differentiability of the inverse function of $F$. While it is true that the theorem can be deduced from the Implicit Function Theorem (and I can trace those back to the 19th century), I would like to know who was the first to formulate a modern version.

By a modern version I mean the following:

[Inverse Function Theorem] Suppose that $\mathbf{f}$ is a function defined on an open $n$-ball $A$, with values in $\mathbb{R}^n$, and that its partial derivatives are continuous in $A$. Let $\mathbf{c}\in A$ and suppose that $\mathbf{D}\mathbf{f}(\mathbf{c})$ is bijective. Then there exists an open $n$-ball $B$ with center $\mathbf{c}$, such that:

(a) the restriction $\mathbf{f}|B$ is a bijection between $B$ and $f(B)$;

(b) the set $V=f(B)$ is open;

(c) the inverse $\mathbf{h}=(\mathbf{f}|B)^{-1}$ is uniformly continuous on $V$;

(d) $\mathbf{h}$ has continuous partial derivatives;

(e) $\mathbf{D}\mathbf{h}(\mathbf{v})=(\mathbf{D}\mathbf{f}(\mathbf{h}(\mathbf{v})))^{-1}$, for $\mathbf{v}\in V$.

I can trace such a statement to Apostol's 1957 "Mathematical Analysis".

Thanks!

a modern version? – Mariano Suárez-Alvarez♦ Apr 17 '12 at 22:59