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:

(1)
\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 $Y$.