TITLE: Inverse/Implicit Function Theorem Reasons?
QUESTION [29 upvotes]: I watched an ICTP lecture on elementary real analysis & the lecturer went to great
pains to emphasize the importance of the intermediate value theorem because it is 
what generalizes to higher dimensions via connectedness & how Bolzano-Weierstrass 
generalizes to metric spaces & how just a few results following from these are what
really matters. Regardless of how true that is (I think he's biased because he is a 
functional analyst (analysist?)) I found that bit of intuition & motivation extremely
clarifying & I'd read a lot of links, wiki's etc... just looking for that kind of 
motivation but it was nowhere to be found beforehand. 
Similarly, I've read a lot about the implicit/inverse function theorems but I just
don't understand the reasons for putting so much focus on them, but that's because 
I don't really know what they are saying & that's partly because I don't understand
what you need to know to really appreciate these theorems.
I guess what I'm asking for is an insightful & human explanation of what these theories
are, what you need to know to lead up to them, why you need to know that certain material & why these things are so powerful (for instance, I believe you can use these to prove the Lagrange Multipliers theorem, though I don't know why it has anything to do with it, I also know that knowing one means you can prove the other & that it doesn't matter which direction you approach from, but again I don't appreciate why that is). 
I'm more interested in the surrounding theory, i.e. what you need to know, why you 
need to know that & why it is so important, than what the theorem says so I'd much
rather prefer to have the motivation so that I could prove this myself. 
(Yes I've read the wiki page, the threads on this site, the many articles apparently trying to motivate it, I just feel I haven't read anything that directly quells my aforementioned concerns so I think that justifies the thread).
Thanks for your time.

REPLY [4 votes]: Suppose $f:\mathbb{R}^m\to\mathbb{R}^n$ is a smooth map. Recall that, at any given point $x \in \mathbb{R}^m,$ the derivative of $f$ is a linear map $D_x f: \mathbb{R}^m \to \mathbb{R}^n$ given by
$$ D_x f(\vec{v}) = \lim_{t\to 0} \frac{f(x + t \vec{v}) - f(x)}{t}.$$
Recall that, even when $f$ is not invertible, for any given point $y \in \mathbb{R}^n,$ we may still define the preimage $f^{-1}(y)$ to be the set of all points $x \in \mathbb{R}^m$ such that $f(x) = y.$ 
We say that a point $y \in \mathbb{R}^n$ is a regular point when for every point $x \in f^{-1}(y),$ $D_x f$ has full rank. In particular, if $m > n,$ this implies that $D_x f$ is surjective. Whenever $D_x f$ is surjective, we say that $f$ is a submersion at $x.$
Local Submersion Theorem: If $f$ is a submersion at $x,$ then there are local coordinates $(\tilde{x}_1,\ldots, \tilde{x}_m)$ around $x$ such that $f(\tilde{x}_1,\ldots,\tilde{x}_m) = (\tilde{x}_1,\ldots,\tilde{x}_n).$
Preimage Theorem: If $y \in \mathbb{R}^n$ is a regular point, then $f^{-1}(y)$ is a smooth $(m-n)$-dimensional submanifold of $\mathbb{R}^m.$ That is, $f^{-1}(y)$ is locally homeomorphic (by a smooth invertible map) to $\mathbb{R}^{(m-n)}.$ That is, up close, $f^{-1}(y)$ looks just like $\mathbb{R}^{(m-n)}.$
The proof of the local submersion theorem depends critically upon IFT. In turn, the proof of the preimage theorem depends critically upon the local submersion theorem.
Transversality and intersection theory---as given in, for instance, Guillemin and Pollack's classic Differential Topology---depend critically upon the preimage theorem. These tools are very useful; they can be used to prove the Borsuk-Ulam theorem and the fundamental theorem of algebra, and they are essential tools for the development of Lefschetz fixed point theory and the nebula of results called the "Poincare-Hopf index theorem." The Poincare-Hopf index theorem, in turn, is used to prove the general Gauss-Bonnet theorem.