Who was the first to formulate the inverse function theorem?

2012-04-17T21:19:23Z

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;

(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!

Comment by john

2012-04-18T18:17:16Z

The answer by Nicola Ciccoli is very nice! I looked up Dini's text and the "Inverse Function Theorem" is almost there. I am not talking about topology, just the fact that in the case of a function $f:\mathbb{R}\to\mathbb{R}$ he explicitly says 'derivate delle funzioni inverse', whereas in the whole 50-page Chapter XIII, he never uses the word inverse. Of course, it is a simple corollary of the Implicit Function Theorem. At some point in time this changed. Now most of the texts prove Inverse Function Theorem first, then derive Implicit Function Theorem. When did that start? Who was the first?

User john - MathOverflow

Who was the first to formulate the inverse function theorem?

Comment by john