Implicit function theorem, theory help.

[Kruum]

For couple of days now I've been trying to figure out the implicit function theorem. Especially why we have to be near a point, why can't we be exactly in the point when trying to find out y as a function of x.

Let's take an easy equation as an example. Find $$y=y(x)$$ when $$f(x,y)=x^2+y^2-1$$. If I solve this for y, it gives me $$y=\sqrt{1-x^2}$$ and $$y=-\sqrt{1-x^2}$$. Now there shouldn't be any reasons for why I can't calculate y, when x=1. But if I use the implicit function theorem, it says you can't define y=y(x) at x=0, because $$f_y(x,y(x))=0$$.

Now ignore the fact that the function in the example is defined explicitly, think we had an implicit function that behaved like the one here. Can anybody explain me why is this?

 

[lippyka]

Why can't we define the function at x=1? What is the reason behind using the implicit function theorem when the equation is defined explicitly.