consider we have a function $f(x,y,z) = x^2 + y^2 + z^2$ then the level surface of $f=1$ can be represented as $z=g(x,y)=\sqrt{1-x^2 - y^2}$ but I read in a book that this is true for $z\neq0$ why this is true only when $z\neq0$?
and another problem I have faced is that according to implicit function theorem there is a small interval such that $(x,y,g(x,y) = (x,y,z)$ why small interval why this isn't always true?