TITLE: Intuitive Explanation of the Inverse Function Theorem
QUESTION [5 upvotes]: What is the intuitive explanation of the Inverse Function Theorem (and generalized to multiple dimensions)?

REPLY [10 votes]: The moral story of all differential calculus is this:

Differentiable functions behave locally like their linearizations.

The degree to which this statement holds can be quantitatively formulated, for example in terms of error estimates for first-order Taylor expansions. It also manifests qualitatively; for an easy example, if $f:\mathbb{R}\to\mathbb{R}$ has a linearization $\ell_p(t) = f(p) + f'(p)(t-p)$ which is increasing (i.e. $f'(p)>0$), then $f$ is also locally increasing.
The moral of the inverse function theorem goes like this. First, let $f$ be differentiable on a neighborhood of $p$, and suppose for simplicity that $f(p) = 0$, so that the linearization of $f$ is $f(x) = D_pf(x-p)$. Then:

If a function $f$ is differentiable on a neighborhood of a point $p$ and its linearization $D_pf$ at $p$ is invertible, then $f$ is locally invertible. Moreover, the inverse of the linearization $(D_pf)^{-1}$ is precisely the linearization of the inverse $f^{-1}$ of $f$ (which we know to be locally defined).

So all we're saying is that invertibility is one of the properties that a function $f$ can copy from its linearization $D_pf$. But because a given linear approximation is only accurate (i.e. has small error) in a small neighborhood of the point of approximation, and you can only expect $f$ to behave like $D_pf$ when the approximation is accurate, you can only conclude that $f$ is invertible on a small neighborhood.