I am really struggling with a theorem in my book (Tao; Analysis II).

I don't think I can properly remember this theorem because it contains so much, plus I dont think I really get the Theorem. So my question to you guys is, can you make this theorem understandable for me. Maybe give some simple examples. Show it's usefulness. Tell me how to remember it, etc.

Any help is really appreciated.

The Implicit Function Theorem:

Let $\displaystyle E$ be an open subset of $\displaystyle \mathbb{R}^n$ and let $\displaystyle f:E\rightarrow \mathbb{R}$ be a continous differentiable function and $\displaystyle y=(y_1,y_2,..,y_n)$ a point in $\displaystyle E$ with $\displaystyle f(y)=0$ and $\displaystyle \frac{\partial f}{\partial x_n}(y)\neq 0$.

Then there exists an open subset $\displaystyle U$ of $\displaystyle \mathbb{R}^{n-1}$ which contains $\displaystyle (y_1,y_2,...,y_{n-1})$ and there exists an open subset $\displaystyle V$ of $\displaystyle E$ that contains $\displaystyle y$ and a function $\displaystyle g:U\rightarrow \mathbb{R}$ such that $\displaystyle g(y_1,...,y_{n-1})=y_n$, and:

$\displaystyle \{(x_1,...,x_n)\in V : f(x_1,...,x_n)=0\}$=$\displaystyle \{(x_1,...,x_{n-1},g(x_1,...,x_{n-1}))x_1,...x_{n-1})\in U\}$

In other words $\displaystyle \{x\in V:f(x)=0\}$ is a graph of a function on $\displaystyle U$. Furthermore $\displaystyle g$ is differentiable in $\displaystyle (y_1,..,y_{n-1})$ and we have:

$\displaystyle \frac{\partial g}{\partial x_j}(y_1,...,y_{n-1})=-\frac{\partial f}{\partial x_j}(y)\slash \frac{\partial f}{\partial x_n}(y)$ for all $\displaystyle 1\leq j \leq n-1$.