I understand the general 'word' definitions of the Implicit Function Theorem and the simple examples such as the one on wikipedia but the version of the Implicit Function theorem given in our lecture notes seems pretty obscure (or perhaps I am just missing something). Here it is.
Let $D \subset \mathbb{R}^m \times \mathbb{R}^p$ be open and $f:D \to \mathbb{R}^n$ be continuously differentiable. Further assume $f(\bar{x},\bar{\lambda}) = 0 \, \, \, \mathrm{and} \, \, \, \det f_1'(\bar{x},\bar{\lambda})\neq 0.$ Then there is a neighbourhood $N$ of $\bar{\lambda}$ and a uniquely determined continuously differentiable mapping $\phi : N \to \mathbb{R}^m \quad N \subset \mathbb{R}^p$ such that for all $\lambda \in N$ $f(\phi(\lambda), \lambda) = 0.$
Can someone please try and explain this particular formulation of the Implicit Function Theorem in a different way? Thanks!