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 be an open subset of and let be a continous differentiable function and a point in with and .

Then there exists an open subset of which contains and there exists an open subset of that contains and a function such that , and:

= x_1,...x_{n-1})\in U\}" alt="\{(x_1,...,x_{n-1},g(x_1,...,x_{n-1}))x_1,...x_{n-1})\in U\}" />

In other words is a graph of a function on . Furthermore is differentiable in and we have:

for all .