The Inverse Function Theorem and the Implicit Function Theorem

In this section, we want to prove the inverse function theorem (which asserts that if a function has invertible differential at a point, then it is locally invertible itself) and the implicit function theorem (which asserts that certain sets are the graphs of functions).

The inverse function theorem

Theorem:

Let $f:\mathbb {R} ^{n}\to \mathbb {R} ^{n}$ be a function which is continuously differentiable in a neighbourhood $x_{0}\in \mathbb {R} ^{n}$ such that $f'(x_{0})$ is invertible. Then there exists an open set $U\subseteq \mathbb {R} ^{n}$ with $x_{0}\in U$ such that $f|_{U}$ is a bijective function with an inverse $f^{-1}:f(U)\to U$ which is differentiable at $x_{0}$ and satisfies

$(f^{-1})'(f(x_{0}))=(f'(x_{0}))^{-1}$.

Proof:

We first reduce to the case $f(x_{0})=0$, $x_{0}=0$ and $f'(x_{0})={\text{Id}}$. Indeed, suppose for all those functions the theorem holds, and let now $h$ be an arbitrary function satisfying the requirements of the theorem (where the differentiability is given at $x_{0}$). We set

${\tilde {h}}(x):=h'(x_{0})^{-1}(h(x_{0}-x)-h(x_{0}))$

and obtain that ${\tilde {h}}$ is differentiable at $0$ with differential ${\text{Id}}$ and ${\tilde {h}}(0)=0$; the first property follows since we multiply both the function and the linear-affine approximation by $h'(x_{0})^{-1}$ and only shift the function, and the second one is seen from inserting $x=0$. Hence, we obtain an inverse of ${\tilde {h}}$ with it's differential at ${\tilde {h}}(0)=0$, and if we now set

$h^{-1}(y):=({\tilde {h}}^{-1}(h'(x_{0})^{-1}(y-h(x_{0})))-x_{0})$,

it can be seen that $h^{-1}$ is an inverse of $h$ with all the required properties (which is a bit of a tedious exercise, but involves nothing more than the definitions).

Thus let $f$ be a function such that $f(0)=0$, $f$ is invertible at $0$ and $f'(0)={\text{Id}}$. We define

$g(x):=f(x)-x$.

The differential of this function is zero (since taking the differential is linear and the differential of the function $x\mapsto x$ is the identity). Since the function $g$ is also continuously differentiable at a small neighbourhood of $0$, we find $\delta >0$ such that

${\frac {\partial g}{\partial x_{j}}}(x)<{\frac {1}{2n^{2}}}$

for all $j\in \{1,\ldots ,n\}$ and $x\in B_{\delta }(0)$. Since further $g(0)=f(0)-0=0$, the general mean-value theorem and Cauchy's inequality imply that for $k\in \{1,\ldots ,n\}$ and $x\in B_{\delta }(0)$,

$|g_{k}(x)|=|\langle x,{\frac {\partial g}{\partial x_{j}}}(t_{k}x)\rangle |\leq \|x\|n{\frac {1}{2n^{2}}}$

for suitable $t_{k}\in [0,1]$. Hence,

$\|g(x)\|\leq |g_{1}(x)|+\cdots +|g_{n}(x)|\leq {\frac {1}{2}}\|x\|$ (triangle inequality),

and thus, we obtain that our preparatory lemma is applicable, and $f$ is a bijection on ${\overline {B_{\delta }(0)}}$, whose image is contained within the open set ${\overline {B_{\delta /2}(0)}}$; thus we may pick Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U := f^{-1}(B_{\delta/2}(0))} , which is open due to the continuity of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} .

Thus, the most important part of the theorem is already done. All that is left to do is to prove differentiability of $f^{-1}$ at $0$. Now we even prove the slightly stronger claim that the differential of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{-1}} at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} is given by the identity, although this would also follow from the chain rule once differentiability is proven.

Note now that the contraction identity for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} implies the following bounds on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2}\|x\| \le \|f(x)\| \le \frac{3}{2} \|x\|} .

The second bound follows from

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|f(x)\| \le \|f(x) - x\| + \|x\| = \|g(x)\| + \|x\| \le \frac{3}{2} \|x\|} ,

and the first bound follows from

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|f(x)\| \ge |\|f(x) - x\| - \|x\|| = \left| \|g(x)\| - \|x\| \right| \ge \frac{1}{2} \|x\|} .

Now for the differentiability at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} . We have, by substitution of limits (as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is continuous and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(0) = 0} ):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \lim_{\mathbf h \to 0} \frac{\|f^{-1}(\mathbf h) - f^{-1}(0) - \operatorname{Id} (\mathbf h - 0)\|}{\|\mathbf h\|} & = \lim_{\mathbf h \to 0} \frac{\|f^{-1}(f(\mathbf h)) - f(\mathbf h)\|}{\|f(\mathbf h)\|} \\ & = \lim_{\mathbf h \to 0} \frac{\|\mathbf h - f(\mathbf h)\|}{\|f(\mathbf h)\|}, \end{align}}

where the last expression converges to zero due to the differentiability of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} with differential the identity, and the sandwhich criterion applied to the expressions

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\|\mathbf h - f(\mathbf h)\|}{\frac{3}{2} \|\mathbf h\|}}

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\|\mathbf h - f(\mathbf h)\|}{\frac{1}{2} \|\mathbf h\|}} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare }

The implicit function theorem

Theorem:

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f: \mathbb R^n \to \mathbb R} be a continuously differentiable function, and consider the set

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S := \{(x_1, \ldots, x_n) \in \mathbb R^n | f(x_1, \ldots, x_n) = 0\}} .

If we are given some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y \in S} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial_n f(y) \neq 0} , then we find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U \subseteq \mathbb R^{n-1}} open with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (y_1, \ldots, y_{n-1}) \in U} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g: U \to S} such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = g(y_1, \ldots, y_{n-1})} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{(z_1, \ldots, z_{n-1}, g(z_1, \ldots, z_{n-1})) | (z_1, \ldots, z_{n-1}) \in U\} \subseteq S} ,

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{(z_1, \ldots, z_{n-1}, g(z_1, \ldots, z_{n-1})) | (z_1, \ldots, z_{n-1}) \in U\}} is open with respect to the subspace topology of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} .

Furthermore, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} is a differentiable function.

Proof:

We define a new function

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F: \mathbb R^n \to \mathbb R^n, F(x_1, \ldots, x_n) := (x_1, \ldots, x_{n-1}, f(x_1, \ldots, x_n))} .

The differential of this function looks like this:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F'(x) = \begin{pmatrix} 1 & 0 & \cdots & & 0 \\ 0 & 1 & & & \vdots \\ \vdots & & \ddots & & \\ 0 & \cdots & 0 & 1 & 0 \\ \partial_1 f(x) & & \cdots & & \partial_n f(x) \end{pmatrix}}

Since we assumed that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial_n f(y) \neq 0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F'(y)} is invertible, and hence the inverse function theorem implies the existence of a small open neighbourhood Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde V \subseteq \mathbb R^n} containing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} such that restricted to that neighbourhood Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is itself invertible, with a differentiable inverse Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F^{-1}} , which is itself defined on an open set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde U} containing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(y)} . Now set first

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U := \{(x_1, \ldots, x_{n-1}) | (x_1, \ldots, x_{n-1}, 0) \in \tilde U \}} ,

which is open with respect to the subspace topology of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb R^{n-1}} , and then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g: U \to \mathbb R, g(x_1, \ldots, x_{n-1}) := \pi_n(F^{-1}(x_1, \ldots, x_{n-1}, 0))} ,

the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -th component of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F^{-1}(x_1, \ldots, x_{n-1}, 0)} . We claim that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} has the desired properties.

Indeed, we first note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F^{-1}(x_1, \ldots, x_{n-1}, 0) = (x_1, \ldots, x_{n-1}, g(x_1, \ldots, x_{n-1}))} , since applying Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} leaves the first Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n-1} components unchanged, and thus we get the identity by observing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(F^{-1}(x)) = x} . Let thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (z_1, \ldots, z_{n-1}) \in U} . Then

${\begin{aligned}f(z_{1},\ldots ,z_{n-1},g(z_{1},\ldots ,z_{n-1}))&=(\pi _{n}\circ F)(F^{-1}(z_{1},\ldots ,z_{n-1},0))\\&=\pi _{n}((F\circ F^{-1})(z_{1},\ldots ,z_{n-1},0))=0\end{aligned}}$.

Furthermore, the set

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{(z_1, \ldots, z_{n-1}, g(z_1, \ldots, z_{n-1})) | (z_1, \ldots, z_{n-1}) \in U\}}

is open with respect to the subspace topology on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} . Indeed, we show

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{(z_1, \ldots, z_{n-1}, g(z_1, \ldots, z_{n-1})) | (z_1, \ldots, z_{n-1}) \in U\} = S \cap \tilde V} .

For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \subseteq} , we first note that the set on the left hand side is in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} , since all points in it are mapped to zero by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} . Further,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(z_1, \ldots, z_{n-1}, g(z_1, \ldots, z_{n-1})) = (z_1, \ldots, z_{n-1}, 0) \in \tilde U}

and hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \subseteq} is completed when applying Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F^{-1}} . For the other direction, let a point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_1, \ldots, x_n)} in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S \cap \tilde V} be given, apply Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} to get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F((x_1, \ldots, x_n)) = (x_1, \ldots, x_{n-1}, 0) \in \tilde U}

and hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_1, \ldots, x_{n-1}) \in U} ; further

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_1, \ldots, x_{n-1}, g(x_1, \ldots, x_{n-1})) = (x_1, \ldots, x_n)}

by applying Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} to both sides of the equation.

Now Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} is automatically differentiable as the component of a differentiable function. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare }

Informally, the above theorem states that given a set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{x \in \mathbb R^n | f(x) = 0\}} , one can choose the first Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n-1} coordinates as a "base" for a function, whose graph is precisely a local bit of that set.

Licensing

Content obtained and/or adapted from:

• Inverse function theorem, implicit function theorem, Wikibooks: Calculus under a CC BY-SA license