The Inverse Function Theorem for Functions from Rn to Rn

The Inverse Function Theorem for Functions from Rn to Rn

On the Nonzero Jacobian Determinants of Differentiable Functions from Rn to Rn page we stated a bunch of important theorems regarding functions $\mathbf{f} : \mathbb{R}^n \to \mathbb{R}^n$ and the property for which the Jacobian determinant of $\mathbf{f}$ is nonzero.

We will make one definition before stating one of the most significant theorems in real analysis.

Definition: Let $A \subseteq \mathbb{R}^n$ be open and let $\mathbf{f} : A \to \mathbb{R}^n$. Then $\mathbf{f}$ is said to be a Continuously Differentiable Function on $A$ or a $C^1$ function on $A$ if the partial derivatives of the Jacobian matrix exist and are continuous on $A$, that is, $D_j f_i (\mathbf{x})$ exist and are continuous for all $\mathbf{x} \in A$ and for all $i, j \in \{ 1, 2, ..., n \}$.

Most functions that we will look at will be continuously differentiable where they are defined - but not all. For example, consider the function $\mathbf{f} : \mathbb{R}^2 \to \mathbb{R}^2$ defined by $\mathbf{f}(x, y) = (e^x \cos y, e^x \sin y)$. Then $f_1(x, y) = e^x \cos y$ and $f_2(x, y) = e^x \sin y$ and:

\begin{align} \quad \frac{\partial f_1}{\partial x} = e^x \cos y \quad , \quad \frac{\partial f_1}{\partial y} = - e^x \sin y \quad , \quad \frac{\partial f_2}{\partial x} = e^x \sin y \quad , \quad \frac{\partial f_2}{\partial y} = e^x \cos y \end{align}

We see that all of these partial derivatives exist and are clearly continuous on $\mathbb{R}^2$, so $\mathbf{f}$ is continuously differentiable on $\mathbb{R}^2$, i.e., $C^1$ on $\mathbb{R}^2$.

We are now ready to state the famous Inverse Function theorem.

Theorem 1 (The Inverse Function Theorem): Let $A \subseteq \mathbb{R}^n$ be open and let $\mathbf{f} : A \to \mathbb{R}^n$ be a $C^1$ function on $A$. If there exists a point $\mathbf{a} \in A$ for which $\mathbf{J}_{\mathbf{f}} (\mathbf{a}) \neq 0$ then there exists open sets $X \subseteq A$ and $Y \subseteq f(A)$ and a function $\mathbf{g} :Y \to X$ for which:
a) $\mathbf{a} \in X$ and $\mathbf{f}(\mathbf{a}) \in Y$.
b) $\mathbf{f}$ is bijective on $X$.
c) $\mathbf{g} (Y) = X$ and $\mathbf{g}(\mathbf{f}(\mathbf{x})) = \mathbf{x}$ for all $\mathbf{x} \in X$.
d) $\mathbf{g}$ is a $C^1$ function on $Y$.
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