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!
Best Answer
More probably than not it will be rather difficult to have a final word on such kind of question, since it depends on how much you insist on having all words completely respected. The question requires tracking the moment in history of math when it became more common to talk about open sets rather than neighbourhoods, when it became more common to talk about bijections rather than insisting that certain coordinates may be expressed as functions of others and so on...
Since Ottem comments point to a paper that refers to U. Dini "Lezioni di Analisi Infinitesimale", dating 1877, let me comment on what you will find here and you will not. (If you read Italian you can find them here http://ebooks.library.cornell.edu/m/math/index.php)
Let me start by remarking that the "Implicit Function Theorem" in Italy is also called Dini's Theorem, since he is credited to be the one giving a rigorous proof, basing on modern standards. His lecture notes of 1887 contain also the Inverse Function Theorem. He does not mention, of course, open sets, but he insists on the fact that results are valid in a small neighbourhood of $x_0$ when the Jacobian determinant is non zero at $x_0$. He does not talk about bijectivity but about the fact the certain functions $y_1,\ldots,y_n$ can be expressed as functions of the original variables $x_1,\ldots,x_n$ and he explicits refer to the fact that this "inverse" has continuous partial derivatives (finite e continue assieme alle loro derivate parziali prime). No reference to uniform continuity, though.
Basically you have everything in here. Now if you want a closer terminology to Apostol 1957 I guess you have to stroll through textbooks going back in time. Since the whole language of topology was developed in the first years of the xxth Century and since one may expect some years to spread it at the level of lecture courses you should understand which were the innovative Analysis textbooks in the 30ies...This has to be done country by country, I mean that I expect the result to be highly country-dependent....
In Italy, for example, due to the extreme popularity of Dini's lecture notes (a new edition of which was published in 1907) also subsequent "Lezioni di Analisi Infinitesimale" were following a similar approach and language (I have checked those of Guido Fubini, dating 1920, since I have them at home).