9.24 Theorem Suppose $\mathbf{f}$ is a $\mathscr{C}^{\prime}$-mapping of an open set $E \subset R^n$ into $R^n, \mathbf{f}^{\prime}(\mathbf{a})$ is invertible for some $\mathbf{a} \in E$, and $\mathbf{b}=\mathbf{f}(\mathbf{a})$. Then(a) there exist open sets $U$ and $V$ in $R^n$ such that $\mathbf{a} \in U, \mathbf{b} \in V, \mathbf{f}$ is one-toone on $U$, and $\mathbf{f}(U)=V$;(b) if $\mathbf{g}$ is the inverse of $\mathbf{f}[$ which exists, by $(a)]$, defined in $V$ by$$\mathbf{g}(\mathbf{f}(\mathbf{x}))=\mathbf{x} \quad(\mathbf{x} \in U)$$then $\mathbf{g} \in \mathscr{C}^{\prime}(V)$.Writing the equation $\mathbf{y}=\mathbf{f}(\mathbf{x})$ in component form, we arrive at the following interpretation of the conclusion of the theorem: The system of $n$ equations$$y_i=f_1\left(x_1, \ldots, x_n\right) \quad(1 \leq i \leq n)$$can be solved for $x_1, \ldots, x_n$ in terms of $y_1, \ldots, y_n$, if we restrict $\mathbf{x}$ and $\mathbf{y}$ to small enough neighborhoods of $\mathbf{a}$ and $\mathbf{b}$; the solutions are unique and continuously differentiable.Proof(a) Put $\mathbf{f}^{\prime}(a)=A$, and choose $\lambda$ so that$$2 \lambda\left\|A^{-1}\right\|=1 \ \ \ \ \ \ \ \ \ \ \ \ (46)$$Since $\mathbf{f}^{\prime}$ is continuous at a, there is an open ball $U \subset E$, with center at a, such that$$\left\|\mathbf{f}^{\prime}(\mathbf{x})-A\right\|<\lambda \quad(\mathbf{x} \in U) . \ \ \ \ \ \ \ \ \ \ \ \ (47)$$We associate to each $y\in \mathbb{R}^n$ a function $\varphi$, defined by$$\varphi(x) = x + A^{-1}(y - f(x)) \ \ \ \ \ \ \ \ \ \ \ \ (48)$$$(x \in E)$.Note that $f(x) = y$ if and only if xis a.fixed point of $\varphi$,.Since $\varphi'(x)$ = I - A^{-1}f'(x) = A^{-1{(A - f'(x))$, (46) and (47) implythat$$\left|\varphi^{\prime}(\mathbf{x})\right|<\frac{1}{2} \quad(\mathbf{x} \in U) \ \ \ \ \ \ \ \ \ \ \ \ (49) .$$ Hence$$\left|\varphi\left(\mathbf{x}_1\right)-\varphi\left(\mathbf{x}_2\right)\right| \leq \frac{1}{2}\left|\mathbf{x}_1-\mathbf{x}_2\right| \quad\left(\mathbf{x}_1, \mathbf{x}_2 \in U\right)\ \ \ \ (50)$$
We associate to each $y\in \mathbb{R}^n$ a function $\varphi$, defined by$$\varphi(x) = x + A^{-1}(y - f(x)) \ \ \ \ \ \ \ \ \ \ \ \ (48)$$$(x \in E)$.
I have so many questions there, what is "y" is it a constant ? or does it depend on $x$? Is it arbitrary? If so why do we even bother to write it down ? If it is not arbitrary what is it ?