I'm a physics major trying to learn more about this theorem, could someone help me with this proof, so I could understand it?
Generalize the proof of the implicit function theorem to get the following result: Let $f$ be a continuous real-valued function on an open subset of $E^{n+1}$ that contains the point $(a_1,...,a_n,b),$ with $f(a_1,...,a_n,b)=0$. Suppose that $\frac{\partial f}{\partial y}(a_1,...,a_n,b) \neq 0.$
Then there exist positive real numbers $h$ and $k$ such that there exists a unique function $\gamma:$ {$(x_1,...,x_n) \in E^n : (x_1-a_1)^2+...+(x_n-a_n)^2 < h^2$} $\rightarrow$ {$y \in \mathbb{R} : |y-b| < k$} such that $f(x_1,...,x_n,\gamma(x_1,...,x_n))=0$ for all $(x_1,...,x_n)$ in question.