inverse function theorem

For functions of a single variable, the theorem states that if $f$ is a continuously differentiable function with nonzero derivative at the point $a$; then $f$ is injective (or bijective onto the image) in a neighborhood of $a$, the inverse is continuously differentiable near $b=f(a)$, and the derivative of the inverse function at $b$ is the reciprocal of the derivative of $f$ at $a$:

$$\backslash bigl(f^\{-1\}\backslash bigr)\text{'}(b)\; =\; \backslash frac\{1\}\{f\text{'}(a)\}\; =\; \backslash frac\{1\}\{f\text{'}(f^\{-1\}(b))\}.$$

It can happen that a function $f$ may be injective near a point $a$ while $f\text{'}(a)\; =\; 0$. An example is $f(x)\; =\; (x\; -\; a)^3$. In fact, for such a function, the inverse cannot be differentiable at $b\; =\; f(a)$, since if $f^\{-1\}$ were differentiable at $b$, then, by the chain rule, $1\; =\; (f^\{-1\}\; \backslash circ\; f)\text{'}(a)\; =\; (f^\{-1\})\text{'}(b)f\text{'}(a)$, which implies $f\text{'}(a)\; \backslash ne\; 0$. (The situation is different for holomorphic functions; see #Holomorphic inverse function theorem below.)

For functions of more than one variable, the theorem states that if $f$ is a continuously differentiable function from an open subset $A$ of $\backslash mathbb\{R\}^n$ into $\backslash R^n$, and the derivative $f\text{'}(a)$ is invertible at a point {{Mvar|a}} (that is, the determinant of the Jacobian matrix of {{Mvar|f}} at {{Mvar|a}} is non-zero), then there exist neighborhoods $U$ of $a$ in $A$ and $V$ of $b\; =\; f(a)$ such that $f(U)\; \backslash subset\; V$ and $f\; :\; U\; \backslash to\; V$ is bijective.Theorem 1.1.7. in {{cite book|title=The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis|series=Classics in Mathematics|first=Lars|last= Hörmander|author-link=Lars Hörmander|publisher=Springer|year= 2015|edition=2nd|

isbn= 978-3-642-61497-2}} Writing $f=(f\_1,\backslash ldots,f\_n)$, this means that the system of {{Mvar|n}} equations $y\_i\; =\; f\_i(x\_1,\; \backslash dots,\; x\_n)$ has a unique solution for $x\_1,\; \backslash dots,\; x\_n$ in terms of $y\_1,\; \backslash dots,\; y\_n$ when $x\; \backslash in\; U,\; y\; \backslash in\; V$. Note that the theorem does not say $f$ is bijective onto the image where $f\text{'}$ is invertible but that it is locally bijective where $f\text{'}$ is invertible.

Moreover, the theorem says that the inverse function $f^\{-1\}\; :\; V\; \backslash to\; U$ is continuously differentiable, and its derivative at $b=f(a)$ is the inverse map of $f\text{'}(a)$; i.e.,

:$(f^\{-1\})\text{'}(b)\; =\; f\text{'}(a)^\{-1\}.$

In other words, if $Jf^\{-1\}(b),\; Jf(a)$ are the Jacobian matrices representing $(f^\{-1\})\text{'}(b),\; f\text{'}(a)$, this means:

:$Jf^\{-1\}(b)\; =\; Jf(a)^\{-1\}.$

The hard part of the theorem is the existence and differentiability of $f^\{-1\}$. Assuming this, the inverse derivative formula follows from the chain rule applied to $f^\{-1\}\backslash circ\; f\; =\; I$. (Indeed, $1=I\text{'}(a)\; =\; (f^\{-1\}\; \backslash circ\; f)\text{'}(a)\; =\; (f^\{-1\})\text{'}(b)\; \backslash circ\; f\text{'}(a).$) Since taking the inverse is infinitely differentiable, the formula for the derivative of the inverse shows that if $f$ is continuously $k$ times differentiable, with invertible derivative at the point {{Mvar|a}}, then the inverse is also continuously $k$ times differentiable. Here $k$ is a positive integer or $\backslash infty$.

There are two variants of the inverse function theorem. Given a continuously differentiable map $f\; :\; U\; \backslash to\; \backslash mathbb\{R\}^m$, the first is

• The derivative $f\text{'}(a)$ is surjective (i.e., the Jacobian matrix representing it has rank $m$) if and only if there exists a continuously differentiable function $g$ on a neighborhood $V$ of $b\; =\; f(a)$ such that $f\; \backslash circ\; g\; =\; I$ near $b$,

and the second is

• The derivative $f\text{'}(a)$ is injective if and only if there exists a continuously differentiable function $g$ on a neighborhood $V$ of $b\; =\; f(a)$ such that $g\; \backslash circ\; f\; =\; I$ near $a$.

In the first case (when $f\text{'}(a)$ is surjective), the point $b\; =\; f(a)$ is called a regular value. Since $m\; =\; \backslash dim\; \backslash ker(f\text{'}(a))\; +\; \backslash dim\; \backslash operatorname\{im\}(f\text{'}(a))$, the first case is equivalent to saying $b\; =\; f(a)$ is not in the image of critical points $a$ (a critical point is a point $a$ such that the kernel of $f\text{'}(a)$ is nonzero). The statement in the first case is a special case of the submersion theorem.

These variants are restatements of the inverse functions theorem. Indeed, in the first case when $f\text{'}(a)$ is surjective, we can find an (injective) linear map $T$ such that $f\text{'}(a)\; \backslash circ\; T\; =\; I$. Define $h(x)\; =\; a\; +\; Tx$ so that we have:

:$(f\; \backslash circ\; h)\text{'}(0)\; =\; f\text{'}(a)\; \backslash circ\; T\; =\; I.$

Thus, by the inverse function theorem, $f\; \backslash circ\; h$ has inverse near $0$; i.e., $f\; \backslash circ\; h\; \backslash circ\; (f\; \backslash circ\; h)^\{-1\}\; =\; I$ near $b$. The second case ($f\text{'}(a)$ is injective) is seen in the similar way.

Consider the vector-valued function $F:\backslash mathbb\{R\}^2\backslash to\backslash mathbb\{R\}^2\backslash !$ defined by:

:$$

F(x,y)=

\begin{bmatrix}

{e^x \cos y}\\

{e^x \sin y}\\

\end{bmatrix}.

The Jacobian matrix of it at $(x,\; y)$ is:

:$$

JF(x,y)=

\begin{bmatrix}

{e^x \cos y} & {-e^x \sin y}\\

{e^x \sin y} & {e^x \cos y}\\

\end{bmatrix}

with the determinant:

:$$

\det JF(x,y)=

e^{2x} \cos^2 y + e^{2x} \sin^2 y=

e^{2x}.

\,\!

The determinant $e^\{2x\}\backslash !$ is nonzero everywhere. Thus the theorem guarantees that, for every point {{Mvar|p}} in $\backslash mathbb\{R\}^2\backslash !$, there exists a neighborhood about {{Mvar|p}} over which {{Mvar|F}} is invertible. This does not mean {{Mvar|F}} is invertible over its entire domain: in this case {{Mvar|F}} is not even injective since it is periodic: $F(x,y)=F(x,y+2\backslash pi)\backslash !$.

File:Inv-Fun-Thm-3.png

If one drops the assumption that the derivative is continuous, the function no longer need be invertible. For example $f(x)\; =\; x\; +\; 2x^2\backslash sin(\backslash tfrac1x)$ and $f(0)=\; 0$ has discontinuous derivative

$f\text{'}\backslash !(x)\; =\; 1\; -2\backslash cos(\backslash tfrac1x)\; +\; 4x\backslash sin(\backslash tfrac1x)$ and $f\text{'}\backslash !(0)\; =\; 1$, which vanishes arbitrarily close to $x=0$. These critical points are local max/min points of $f$, so $f$ is not one-to-one (and not invertible) on any interval containing $x=0$. Intuitively, the slope $f\text{'}\backslash !(0)=1$ does not propagate to nearby points, where the slopes are governed by a weak but rapid oscillation.

As an important result, the inverse function theorem has been given numerous proofs. The proof most commonly seen in textbooks relies on the contraction mapping principle, also known as the Banach fixed-point theorem (which can also be used as the key step in the proof of existence and uniqueness of solutions to ordinary differential equations).{{cite book |first=Robert C. |last=McOwen |title=Partial Differential Equations: Methods and Applications |location=Upper Saddle River, NJ |publisher=Prentice Hall |year=1996 |isbn=0-13-121880-8 |pages=218–224 |chapter=Calculus of Maps between Banach Spaces |chapter-url=https://books.google.com/books?id=TuNHsNC1Yf0C&pg=PA218 }}{{Cite web |url=https://terrytao.wordpress.com/2011/09/12/the-inverse-function-theorem-for-everywhere-differentiable-maps/ |first=Terence |last=Tao |author-link=Terence Tao |title=The inverse function theorem for everywhere differentiable maps |date=September 12, 2011 |access-date=2019-07-26 }}

Since the fixed point theorem applies in infinite-dimensional (Banach space) settings, this proof generalizes immediately to the infinite-dimensional version of the inverse function theorem{{Cite web|url=https://r-grande.github.io/Expository/Inverse%20Function%20Theorem.pdf |title=Inverse Function Theorem|last=Jaffe|first=Ethan}} (see Generalizations below).

An alternate proof in finite dimensions hinges on the extreme value theorem for functions on a compact set.{{harvnb|Spivak|1965|loc=pages 31–35 }} This approach has an advantage that the proof generalizes to a situation where there is no Cauchy completeness (see {{section link||Over_a_real_closed_field}}).

Yet another proof uses Newton's method, which has the advantage of providing an effective version of the theorem: bounds on the derivative of the function imply an estimate of the size of the neighborhood on which the function is invertible.{{cite book |first1=John H. |last1=Hubbard |author-link=John H. Hubbard |first2=Barbara Burke |last2=Hubbard|author2-link=Barbara Burke Hubbard |title=Vector Analysis, Linear Algebra, and Differential Forms: A Unified Approach |edition=Matrix |year=2001 }}

We want to prove the following: Let $D\; \backslash subseteq\; \backslash R$ be an open set with $x\_0\; \backslash in\; D,\; f:\; D\; \backslash to\; \backslash R$ a continuously differentiable function defined on $D$, and suppose that $f\text{'}(x\_0)\; \backslash ne\; 0$. Then there exists an open interval $I$ with $x\_0\; \backslash in\; I$ such that $f$ maps $I$ bijectively onto the open interval $J\; =\; f(I)$, and such that the inverse function $f^\{-1\}\; :\; J\; \backslash to\; I$ is continuously differentiable, and for any $y\; \backslash in\; J$, if $x\; \backslash in\; I$ is such that $f(x)\; =\; y$, then $(f^\{-1\})\text{'}(y)\; =\; \backslash dfrac\{1\}\{f\text{'}(x)\}$.

We may without loss of generality assume that $f\text{'}(x\_0)\; >\; 0$. Given that $D$ is an open set and $f\text{'}$ is continuous at $x\_0$, there exists $r\; >\; 0$ such that $(x\_0\; -\; r,\; x\_0\; +\; r)\; \backslash subseteq\; D$ and

In particular,

This shows that $f$ is strictly increasing for all . Let $\backslash delta\; >\; 0$ be such that . Then $[x\; -\; \backslash delta,\; x\; +\; \backslash delta]\; \backslash subseteq\; (x\_0\; -\; r,\; x\_0\; +\; r)$. By the intermediate value theorem, we find that $f$ maps the interval $[x\; -\; \backslash delta,\; x\; +\; \backslash delta]$ bijectively onto $[f(x\; -\; \backslash delta),\; f(x\; +\; \backslash delta)]$. Denote by $I\; =\; (x-\backslash delta,\; x+\backslash delta)$ and $J\; =\; (f(x\; -\; \backslash delta),f(x\; +\; \backslash delta))$. Then $f:\; I\; \backslash to\; J$ is a bijection and the inverse $f^\{-1\}:\; J\; \backslash to\; I$ exists. The fact that $f^\{-1\}:\; J\; \backslash to\; I$ is differentiable follows from the differentiability of $f$. In particular, the result follows from the fact that if $f:\; I\; \backslash to\; \backslash R$ is a strictly monotonic and continuous function that is differentiable at $x\_0\; \backslash in\; I$ with $f\text{'}(x\_0)\; \backslash ne\; 0$, then $f^\{-1\}:\; f(I)\; \backslash to\; \backslash R$ is differentiable with $(f^\{-1\})\text{'}(y\_0)\; =\; \backslash dfrac\{1\}\{f\text{'}(y\_0)\}$, where $y\_0\; =\; f(x\_0)$ (a standard result in analysis). This completes the proof.

To prove existence, it can be assumed after an affine transformation that $f(0)=0$ and $f^\backslash prime(0)=I$, so that $a=b=0$.

By the mean value theorem for vector-valued functions, for a differentiable function $u:[0,1]\backslash to\backslash mathbb\; R^m$, $\backslash |u(1)-u(0)\backslash |\backslash le\; \backslash sup\_\{0\backslash le\; t\backslash le\; 1\}\; \backslash |u^\backslash prime(t)\backslash |$. Setting $u(t)=f(x+t(x^\backslash prime\; -x))\; -\; x-t(x^\backslash prime-x)$, it follows that

:$\backslash |f(x)\; -\; f(x^\backslash prime)\; -\; x\; +\; x^\backslash prime\backslash |\; \backslash le\; \backslash |x\; -x^\backslash prime\backslash |\backslash ,\backslash sup\_\{0\backslash le\; t\; \backslash le\; 1\}\; \backslash |f^\backslash prime(x+t(x^\backslash prime\; -x))-I\backslash |.$

Now choose $\backslash delta>0$ so that for . Suppose that and define $x\_n$ inductively by $x\_0=0$ and $x\_\{n+1\}=x\_n\; +\; y\; -\; f(x\_n)$. The assumptions show that if then

:$\backslash |f(x)-f(x^\backslash prime)\; -\; x\; +\; x^\backslash prime\backslash |\; \backslash le\; \backslash |x-x^\backslash prime\backslash |/2$.

In particular $f(x)=f(x^\backslash prime)$ implies $x=x^\backslash prime$. In the inductive scheme

and . Thus $(x\_n)$ is a Cauchy sequence tending to $x$. By construction $f(x)=y$ as required.

To check that $g=f^\{-1\}$ is C¹, write $g(y+k)\; =\; x+h$ so that

$f(x+h)=f(x)+k$. By the inequalities above, so that .

On the other hand, if $A=f^\backslash prime(x)$, then . Using the geometric series for $B=I-A$, it follows that . But then

:$\{\backslash |g(y+k)\; -g(y)\; -\; f^\backslash prime(g(y))^\{-1\}k\; \backslash |\; \backslash over\; \backslash |k\backslash |\}$

= {\|h -f^\prime(x)^{-1}[f(x+h)-f(x)]\| \over \|k\|}

\le 4 {\|f(x+h) - f(x) -f^\prime(x)h\|\over \|h\|}

tends to 0 as $k$ and $h$ tend to 0, proving that $g$ is C¹ with $g^\backslash prime(y)=f^\backslash prime(g(y))^\{-1\}$.

The proof above is presented for a finite-dimensional space, but applies equally well for Banach spaces. If an invertible function $f$ is C^k with $k>1$, then so too is its inverse. This follows by induction using the fact that the map $F(A)=A^\{-1\}$ on operators is C^k for any $k$ (in the finite-dimensional case this is an elementary fact because the inverse of a matrix is given as the adjugate matrix divided by its determinant).

{{cite book|title=Calcul Differentiel|language=fr|first=Henri|last= Cartan|author-link= Henri Cartan|publisher=Hermann|year= 1971|isbn=978-0-395-12033-0 |pages=55–61}} The method of proof here can be found in the books of Henri Cartan, Jean Dieudonné, Serge Lang, Roger Godement and Lars Hörmander.

Here is a proof based on the contraction mapping theorem. Specifically, following T. Tao,Theorem 17.7.2 in {{cite book|mr=3310023|last1=Tao|first1=Terence|title=Analysis. II|edition=Third edition of 2006 original|series=Texts and Readings in Mathematics|volume=38|publisher=Hindustan Book Agency|location=New Delhi|year=2014|isbn=978-93-80250-65-6|zbl=1300.26003}} it uses the following consequence of the contraction mapping theorem.

{{math_theorem|name=Lemma|math_statement=Let $B(0,\; r)$ denote an open ball of radius r in $\backslash mathbb\{R\}^n$ with center 0 and $g\; :\; B(0,\; r)\; \backslash to\; \backslash mathbb\{R\}^n$ a map with a constant such that

:$|g(y)\; -\; g(x)|\; \backslash le\; c|y-x|$

for all $x,\; y$ in $B(0,\; r)$. Then for $f\; =\; I\; +\; g$ on $B(0,\; r)$, we have

:$(1-c)|x\; -\; y|\; \backslash le\; |f(x)\; -\; f(y)|,$

in particular, f is injective. If, moreover, $g(0)\; =\; 0$, then

:$B(0,\; (1-c)r)\; \backslash subset\; f(B(0,\; r))\; \backslash subset\; B(0,\; (1+c)r)$.

More generally, the statement remains true if $\backslash mathbb\{R\}^n$ is replaced by a Banach space. Also, the first part of the lemma is true for any normed space.}}

Basically, the lemma says that a small perturbation of the identity map by a contraction map is injective and preserves a ball in some sense. Assuming the lemma for a moment, we prove the theorem first. As in the above proof, it is enough to prove the special case when $a\; =\; 0,\; b\; =\; f(a)\; =\; 0$ and $f\text{'}(0)\; =\; I$. Let $g\; =\; f\; -\; I$. The mean value inequality applied to $t\; \backslash mapsto\; g(x\; +\; t(y\; -\; x))$ says:

:

Since $g\text{'}(0)\; =\; I\; -\; I\; =\; 0$ and $g\text{'}$ is continuous, we can find an $r\; >\; 0$ such that

:$|g(y)\; -\; g(x)|\; \backslash le\; 2^\{-1\}|y-x|$

for all $x,\; y$ in $B(0,\; r)$. Then the early lemma says that $f\; =\; g\; +\; I$ is injective on $B(0,\; r)$ and $B(0,\; r/2)\; \backslash subset\; f(B(0,\; r))$. Then

:$f\; :\; U\; =\; B(0,\; r)\; \backslash cap\; f^\{-1\}(B(0,\; r/2))\; \backslash to\; V\; =\; B(0,\; r/2)$

is bijective and thus has an inverse. Next, we show the inverse $f^\{-1\}$ is continuously differentiable (this part of the argument is the same as that in the previous proof). This time, let $g\; =\; f^\{-1\}$ denote the inverse of $f$ and $A\; =\; f\text{'}(x)$. For $x\; =\; g(y)$, we write $g(y\; +\; k)\; =\; x\; +\; h$ or $y\; +\; k\; =\; f(x+h)$. Now, by the early estimate, we have

:$|h\; -\; k|\; =\; |f(x+h)\; -\; f(x)\; -\; h|\; \backslash le\; |h|/2$

and so $|h|/2\; \backslash le\; |k|$. Writing $\backslash |\; \backslash cdot\; \backslash |$ for the operator norm,

:$|g(y+k)\; -\; g(y)\; -\; A^\{-1\}\; k|\; =\; |h\; -\; A^\{-1\}(f(x\; +\; h)\; -\; f(x))|\; \backslash le\; \backslash |A^\{-1\}\backslash ||Ah\; -\; f(x+h)\; +\; f(x)|.$

As $k\; \backslash to\; 0$, we have $h\; \backslash to\; 0$ and $|h|/|k|$ is bounded. Hence, $g$ is differentiable at $y$ with the derivative $g\text{'}(y)\; =\; f\text{'}(g(y))^\{-1\}$. Also, $g\text{'}$ is the same as the composition $\backslash iota\; \backslash circ\; f\text{'}\; \backslash circ\; g$ where $\backslash iota\; :\; T\; \backslash mapsto\; T^\{-1\}$; so $g\text{'}$ is continuous.

It remains to show the lemma. First, we have:

:$|x\; -\; y|\; -\; |f(x)\; -\; f(y)|\; \backslash le\; |g(x)\; -\; g(y)|\; \backslash le\; c|x\; -\; y|,$

which is to say

:$(1\; -\; c)|x\; -\; y|\; \backslash le\; |f(x)\; -\; f(y)|.$

This proves the first part. Next, we show $f(B(0,\; r))\; \backslash supset\; B(0,\; (1-c)r)$. The idea is to note that this is equivalent to, given a point $y$ in $B(0,\; (1-c)\; r)$, find a fixed point of the map

:$F\; :\; \backslash overline\{B\}(0,\; r\text{'})\; \backslash to\; \backslash overline\{B\}(0,\; r\text{'}),\; \backslash ,\; x\; \backslash mapsto\; y\; -\; g(x)$

where such that $|y|\; \backslash le\; (1-c)r\text{'}$ and the bar means a closed ball. To find a fixed point, we use the contraction mapping theorem and checking that $F$ is a well-defined strict-contraction mapping is straightforward. Finally, we have: $f(B(0,\; r))\; \backslash subset\; B(0,\; (1+c)r)$ since

:$|f(x)|\; =\; |x\; +\; g(x)\; -\; g(0)|\; \backslash le\; (1+c)|x|.\; \backslash square$

As might be clear, this proof is not substantially different from the previous one, as the proof of the contraction mapping theorem is by successive approximation.

The inverse function theorem can be used to solve a system of equations

:$\backslash begin\{align\}$

&f_1(x) = y_1 \\

&\quad \vdots\\

&f_n(x) = y_n,\end{align}

i.e., expressing $y\_1,\; \backslash dots,\; y\_n$ as functions of $x\; =\; (x\_1,\; \backslash dots,\; x\_n)$, provided the Jacobian matrix is invertible. The implicit function theorem allows to solve a more general system of equations:

:$\backslash begin\{align\}$

&f_1(x, y) = 0 \\

&\quad \vdots\\

&f_n(x, y) = 0\end{align}

for $y$ in terms of $x$. Though more general, the theorem is actually a consequence of the inverse function theorem. First, the precise statement of the implicit function theorem is as follows:{{harvnb|Spivak|1965|loc=Theorem 2-12.}}

• given a map $f\; :\; \backslash mathbb\{R\}^n\; \backslash times\; \backslash mathbb\{R\}^m\; \backslash to\; \backslash mathbb\{R\}^m$, if $f(a,\; b)\; =\; 0$, $f$ is continuously differentiable in a neighborhood of $(a,\; b)$ and the derivative of $y\; \backslash mapsto\; f(a,\; y)$ at $b$ is invertible, then there exists a differentiable map $g\; :\; U\; \backslash to\; V$ for some neighborhoods $U,\; V$ of $a,\; b$ such that $f(x,\; g(x))\; =\; 0$. Moreover, if $f(x,\; y)\; =\; 0,\; x\; \backslash in\; U,\; y\; \backslash in\; V$, then $y\; =\; g(x)$; i.e., $g(x)$ is a unique solution.

To see this, consider the map $F(x,\; y)\; =\; (x,\; f(x,\; y))$. By the inverse function theorem, $F\; :\; U\; \backslash times\; V\; \backslash to\; W$ has the inverse $G$ for some neighborhoods $U,\; V,\; W$. We then have:

:$(x,\; y)\; =\; F(G\_1(x,\; y),\; G\_2(x,\; y))\; =\; (G\_1(x,\; y),\; f(G\_1(x,\; y),\; G\_2(x,\; y))),$

implying $x\; =\; G\_1(x,\; y)$ and $y\; =\; f(x,\; G\_2(x,\; y)).$ Thus $g(x)\; =\; G\_2(x,\; 0)$ has the required property. $\backslash square$

In differential geometry, the inverse function theorem is used to show that the pre-image of a regular value under a smooth map is a manifold.{{harvnb|Spivak|1965|loc=Theorem 5-1. and Theorem 2-13.}} Indeed, let $f\; :\; U\; \backslash to\; \backslash mathbb\{R\}^r$ be such a smooth map from an open subset of $\backslash mathbb\{R\}^n$ (since the result is local, there is no loss of generality with considering such a map). Fix a point $a$ in $f^\{-1\}(b)$ and then, by permuting the coordinates on $\backslash mathbb\{R\}^n$, assume the matrix $\backslash left\; [\; \backslash frac\{\backslash partial\; f\_i\}\{\backslash partial\; x\_j\}(a)\; \backslash right]\_\{1\; \backslash le\; i,\; j\; \backslash le\; r\}$ has rank $r$. Then the map $F\; :\; U\; \backslash to\; \backslash mathbb\{R\}^r\; \backslash times\; \backslash mathbb\{R\}^\{n-r\}\; =\; \backslash mathbb\{R\}^n,\; \backslash ,\; x\; \backslash mapsto\; (f(x),\; x\_\{r+1\},\; \backslash dots,\; x\_n)$ is such that $F\text{'}(a)$ has rank $n$. Hence, by the inverse function theorem, we find the smooth inverse $G$ of $F$ defined in a neighborhood $V\; \backslash times\; W$ of $(b,\; a\_\{r+1\},\; \backslash dots,\; a\_n)$. We then have

:$x\; =\; (F\; \backslash circ\; G)(x)\; =\; (f(G(x)),\; G\_\{r+1\}(x),\; \backslash dots,\; G\_n(x)),$

which implies

:$(f\; \backslash circ\; G)(x\_1,\; \backslash dots,\; x\_n)\; =\; (x\_1,\; \backslash dots,\; x\_r).$

That is, after the change of coordinates by $G$, $f$ is a coordinate projection (this fact is known as the submersion theorem). Moreover, since $G\; :\; V\; \backslash times\; W\; \backslash to\; U\text{'}\; =\; G(V\; \backslash times\; W)$ is bijective, the map

:$g\; =\; G(b,\; \backslash cdot)\; :\; W\; \backslash to\; f^\{-1\}(b)\; \backslash cap\; U\text{'},\; \backslash ,\; (x\_\{r+1\},\; \backslash dots,\; x\_n)\; \backslash mapsto\; G(b,\; x\_\{r+1\},\; \backslash dots,\; x\_n)$

is bijective with the smooth inverse. That is to say, $g$ gives a local parametrization of $f^\{-1\}(b)$ around $a$. Hence, $f^\{-1\}(b)$ is a manifold. $\backslash square$ (Note the proof is quite similar to the proof of the implicit function theorem and, in fact, the implicit function theorem can be also used instead.)

More generally, the theorem shows that if a smooth map $f\; :\; P\; \backslash to\; E$ is transversal to a submanifold $M\; \backslash subset\; E$, then the pre-image $f^\{-1\}(M)\; \backslash hookrightarrow\; P$ is a submanifold.{{cite web|website=northwestern.edu|title=Transversality

|url=https://sites.math.northwestern.edu/~jnkf/classes/mflds/4transversality.pdf}}

The inverse function theorem is a local result; it applies to each point. A priori, the theorem thus only shows the function $f$ is locally bijective (or locally diffeomorphic of some class). The next topological lemma can be used to upgrade local injectivity to injectivity that is global to some extent.

{{math_theorem|name=Lemma|math_statement=One of Spivak's books (Editorial note: give the exact location).{{full citation needed|date=August 2023}}{{harvnb|Hirsch|1976|loc=Ch. 2, § 1., Exercise 7.}} NB: This one is for a $C^1$-immersion. If $A$ is a closed subset of a (second-countable) topological manifold $X$ (or, more generally, a topological space admitting an exhaustion by compact subsets) and $f\; :\; X\; \backslash to\; Z$, $Z$ some topological space, is a local homeomorphism that is injective on $A$, then $f$ is injective on some neighborhood of $A$.}}

Proof:Lemma 13.3.3. of [https://www.utsc.utoronto.ca/people/kupers/wp-content/uploads/sites/50/2020/12/difffop-2020.pdf Lectures on differential topology] utoronto.ca First assume $X$ is compact. If the conclusion of the theorem is false, we can find two sequences $x\_i\; \backslash ne\; y\_i$ such that $f(x\_i)\; =\; f(y\_i)$ and $x\_i,\; y\_i$ each converge to some points $x,\; y$ in $A$. Since $f$ is injective on $A$, $x\; =\; y$. Now, if $i$ is large enough, $x\_i,\; y\_i$ are in a neighborhood of $x\; =\; y$ where $f$ is injective; thus, $x\_i\; =\; y\_i$, a contradiction.

In general, consider the set $E\; =\; \backslash \{\; (x,\; y)\; \backslash in\; X^2\; \backslash mid\; x\; \backslash ne\; y,\; f(x)\; =\; f(y)\; \backslash \}$. It is disjoint from $S\; \backslash times\; S$ for any subset $S\; \backslash subset\; X$ where $f$ is injective. Let $X\_1\; \backslash subset\; X\_2\; \backslash subset\; \backslash cdots$ be an increasing sequence of compact subsets with union $X$ and with $X\_i$ contained in the interior of $X\_\{i+1\}$. Then, by the first part of the proof, for each $i$, we can find a neighborhood $U\_i$ of $A\; \backslash cap\; X\_i$ such that $U\_i^2\; \backslash subset\; X^2\; -\; E$. Then $U\; =\; \backslash bigcup\_i\; U\_i$ has the required property. $\backslash square$ (See also Dan Ramras (https://mathoverflow.net/users/4042/dan-ramras), On a proof of the existence of tubular neighborhoods., URL (version: 2017-04-13): https://mathoverflow.net/q/58124 for an alternative approach.)

The lemma implies the following (a sort of) global version of the inverse function theorem:

{{math_theorem|name=Inverse function theorem|math_statement=Ch. I., § 3, Exercise 10. and § 8, Exercise 14. in V. Guillemin, A. Pollack. "Differential Topology". Prentice-Hall Inc., 1974. ISBN 0-13-212605-2. Let $f\; :\; U\; \backslash to\; V$ be a map between open subsets of $\backslash mathbb\{R\}^n$ or more generally of manifolds. Assume $f$ is continuously differentiable (or is $C^k$). If $f$ is injective on a closed subset $A\; \backslash subset\; U$ and if the Jacobian matrix of $f$ is invertible at each point of $A$, then $f$ is injective on a neighborhood $A\text{'}$ of $A$ and $f^\{-1\}\; :\; f(A\text{'})\; \backslash to\; A\text{'}$ is continuously differentiable (or is $C^k$).}}

Note that if $A$ is a point, then the above is the usual inverse function theorem.

There is a version of the inverse function theorem for holomorphic maps.

{{math_theorem|name=Theorem|math_statement={{harvnb|Griffiths|Harris|1978|loc=p. 18.}}{{cite book |first1=K. |last1=Fritzsche |first2=H. |last2=Grauert |title=From Holomorphic Functions to Complex Manifolds |publisher=Springer |year=2002 |pages=33–36 |isbn=978-0-387-95395-3 |url=https://books.google.com/books?id=jSeRz36zXIMC&pg=PA33 }} Let $U,\; V\; \backslash subset\; \backslash mathbb\{C\}^n$ be open subsets such that $0\; \backslash in\; U$ and $f\; :\; U\; \backslash to\; V$ a holomorphic map whose Jacobian matrix in variables $z\_i,\; \backslash overline\{z\}\_i$ is invertible (the determinant is nonzero) at $0$. Then $f$ is injective in some neighborhood $W$ of $0$ and the inverse $f^\{-1\}\; :\; f(W)\; \backslash to\; W$ is holomorphic.}}

The theorem follows from the usual inverse function theorem. Indeed, let $J\_\{\backslash mathbb\{R\}\}(f)$ denote the Jacobian matrix of $f$ in variables $x\_i,\; y\_i$ and $J(f)$ for that in $z\_j,\; \backslash overline\{z\}\_j$. Then we have $\backslash det\; J\_\{\backslash mathbb\{R\}\}(f)\; =\; |\backslash det\; J(f)|^2$, which is nonzero by assumption. Hence, by the usual inverse function theorem, $f$ is injective near $0$ with continuously differentiable inverse. By chain rule, with $w\; =\; f(z)$,

:$\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash overline\{z\}\_j\}\; (f\_j^\{-1\}\; \backslash circ\; f)(z)\; =\; \backslash sum\_k\; \backslash frac\{\backslash partial\; f\_j^\{-1\}\}\{\backslash partial\; w\_k\}(w)\; \backslash frac\{\backslash partial\; f\_k\}\{\backslash partial\; \backslash overline\{z\}\_j\}(z)\; +\; \backslash sum\_k\; \backslash frac\{\backslash partial\; f\_j^\{-1\}\}\{\backslash partial\; \backslash overline\{w\}\_k\}(w)\; \backslash frac\{\backslash partial\; \backslash overline\{f\}\_k\}\{\backslash partial\; \backslash overline\{z\}\_j\}(z)$

where the left-hand side and the first term on the right vanish since $f\_j^\{-1\}\; \backslash circ\; f$ and $f\_k$ are holomorphic. Thus, $\backslash frac\{\backslash partial\; f\_j^\{-1\}\}\{\backslash partial\; \backslash overline\{w\}\_k\}(w)\; =\; 0$ for each $k$. $\backslash square$

Similarly, there is the implicit function theorem for holomorphic functions.{{harvnb|Griffiths|Harris|1978|loc=p. 19.}}

As already noted earlier, it can happen that an injective smooth function has the inverse that is not smooth (e.g., $f(x)\; =\; x^3$ in a real variable). This is not the case for holomorphic functions because of:

{{math_theorem|name=Proposition|math_statement= If $f\; :\; U\; \backslash to\; V$ is an injective holomorphic map between open subsets of $\backslash mathbb\{C\}^n$, then $f^\{-1\}\; :\; f(U)\; \backslash to\; U$ is holomorphic.}}

The inverse function theorem can be rephrased in terms of differentiable maps between differentiable manifolds. In this context the theorem states that for a differentiable map $F:\; M\; \backslash to\; N$ (of class $C^1$), if the differential of $F$,

:$dF\_p:\; T\_p\; M\; \backslash to\; T\_\{F(p)\}\; N$

is a linear isomorphism at a point $p$ in $M$ then there exists an open neighborhood $U$ of $p$ such that

:$F|\_U:\; U\; \backslash to\; F(U)$

is a diffeomorphism. Note that this implies that the connected components of {{Mvar|M}} and {{Mvar|N}} containing p and F(p) have the same dimension, as is already directly implied from the assumption that dF_p is an isomorphism.

If the derivative of {{Mvar|F}} is an isomorphism at all points {{Mvar|p}} in {{Mvar|M}} then the map {{Mvar|F}} is a local diffeomorphism.

The inverse function theorem can also be generalized to differentiable maps between Banach spaces {{Mvar|X}} and {{Mvar|Y}}.{{cite book |first=David G. |last=Luenberger |author-link=David Luenberger |title=Optimization by Vector Space Methods |location=New York |publisher=John Wiley & Sons |year=1969 |isbn=0-471-55359-X |pages=240–242 |url=https://books.google.com/books?id=lZU0CAH4RccC&pg=PA240 }} Let {{Mvar|U}} be an open neighbourhood of the origin in {{Mvar|X}} and $F:\; U\; \backslash to\; Y\backslash !$ a continuously differentiable function, and assume that the Fréchet derivative $dF\_0:\; X\; \backslash to\; Y\backslash !$ of {{Mvar|F}} at 0 is a bounded linear isomorphism of {{Mvar|X}} onto {{Mvar|Y}}. Then there exists an open neighbourhood {{Mvar|V}} of $F(0)\backslash !$ in {{Mvar|Y}} and a continuously differentiable map $G:\; V\; \backslash to\; X\backslash !$ such that $F(G(y))\; =\; y$ for all {{Mvar|y}} in {{Mvar|V}}. Moreover, $G(y)\backslash !$ is the only sufficiently small solution {{Mvar|x}} of the equation $F(x)\; =\; y\backslash !$.

There is also the inverse function theorem for Banach manifolds.{{cite book |first=Serge |last=Lang |author-link=Serge Lang |title=Differential Manifolds |location=New York |publisher=Springer |year=1985 |isbn=0-387-96113-5 |pages=13–19 }}

The inverse function theorem (and the implicit function theorem) can be seen as a special case of the constant rank theorem, which states that a smooth map with constant rank near a point can be put in a particular normal form near that point.{{cite book |first=William M. |last=Boothby |title=An Introduction to Differentiable Manifolds and Riemannian Geometry |url=https://archive.org/details/introductiontodi0000boot |url-access=registration |edition=Second |year=1986 |publisher=Academic Press |location=Orlando |isbn=0-12-116052-1 |pages=[https://archive.org/details/introductiontodi0000boot/page/46 46–50] }} Specifically, if $F:M\backslash to\; N$ has constant rank near a point $p\backslash in\; M\backslash !$, then there are open neighborhoods {{Mvar|U}} of {{Mvar|p}} and {{Mvar|V}} of $F(p)\backslash !$ and there are diffeomorphisms $u:T\_pM\backslash to\; U\backslash !$ and $v:T\_\{F(p)\}N\backslash to\; V\backslash !$ such that $F(U)\backslash subseteq\; V\backslash !$ and such that the derivative $dF\_p:T\_pM\backslash to\; T\_\{F(p)\}N\backslash !$ is equal to $v^\{-1\}\backslash circ\; F\backslash circ\; u\backslash !$. That is, {{Mvar|F}} "looks like" its derivative near {{Mvar|p}}. The set of points $p\backslash in\; M$ such that the rank is constant in a neighborhood of $p$ is an open dense subset of {{Mvar|M}}; this is a consequence of semicontinuity of the rank function. Thus the constant rank theorem applies to a generic point of the domain.

When the derivative of {{Mvar|F}} is injective (resp. surjective) at a point {{Mvar|p}}, it is also injective (resp. surjective) in a neighborhood of {{Mvar|p}}, and hence the rank of {{Mvar|F}} is constant on that neighborhood, and the constant rank theorem applies.

If it is true, the Jacobian conjecture would be a variant of the inverse function theorem for polynomials. It states that if a vector-valued polynomial function has a Jacobian determinant that is an invertible polynomial (that is a nonzero constant), then it has an inverse that is also a polynomial function. It is unknown whether this is true or false, even in the case of two variables. This is a major open problem in the theory of polynomials.

When $f:\; \backslash mathbb\{R\}^n\; \backslash to\; \backslash mathbb\{R\}^m$ with $m\backslash leq\; n$, $f$ is $k$ times continuously differentiable, and the Jacobian $A=\backslash nabla\; f(\backslash overline\{x\})$ at a point $\backslash overline\{x\}$ is of rank $m$, the inverse of $f$ may not be unique. However, there exists a local selection function $s$ such that $f(s(y))\; =\; y$ for all $y$ in a neighborhood of $\backslash overline\{y\}\; =\; f(\backslash overline\{x\})$, $s(\backslash overline\{y\})\; =\; \backslash overline\{x\}$, $s$ is $k$ times continuously differentiable in this neighborhood, and $\backslash nabla\; s(\backslash overline\{y\})\; =\; A^T(A\; A^T)^\{-1\}$ ($\backslash nabla\; s(\backslash overline\{y\})$ is the Moore–Penrose pseudoinverse of $A$).{{cite book |last1=Dontchev |first1=Asen L. |last2=Rockafellar |first2=R. Tyrrell |title=Implicit Functions and Solution Mappings: A View from Variational Analysis |date=2014 |publisher=Springer-Verlag |location=New York |isbn=978-1-4939-1036-6 |page=54 |edition=Second}}

The inverse function theorem also holds over a real closed field k (or an O-minimal structure).Theorem 2.11. in {{cite book |doi=10.1017/CBO9780511525919|title=Tame Topology and O-minimal Structures. London Mathematical Society lecture note series, no. 248|year=1998 |last1=Dries |first1=L. P. D. van den |authorlink = Lou van den Dries|isbn=9780521598385|publisher=Cambridge University Press|location=Cambridge, New York, and Oakleigh, Victoria }} Precisely, the theorem holds for a semialgebraic (or definable) map between open subsets of $k^n$ that is continuously differentiable.

The usual proof of the IFT uses Banach's fixed point theorem, which relies on the Cauchy completeness. That part of the argument is replaced by the use of the extreme value theorem, which does not need completeness. Explicitly, in {{section link||A_proof_using_the_contraction_mapping_principle}}, the Cauchy completeness is used only to establish the inclusion $B(0,\; r/2)\; \backslash subset\; f(B(0,\; r))$. Here, we shall directly show $B(0,\; r/4)\; \backslash subset\; f(B(0,\; r))$ instead (which is enough). Given a point $y$ in $B(0,\; r/4)$, consider the function $P(x)\; =\; |f(x)\; -\; y|^2$ defined on a neighborhood of $\backslash overline\{B\}(0,\; r)$. If $P\text{'}(x)\; =\; 0$, then $0\; =\; P\text{'}(x)\; =\; 2[f\_1(x)\; -\; y\_1\; \backslash cdots\; f\_n(x)\; -\; y\_n]f\text{'}(x)$ and so $f(x)\; =\; y$, since $f\text{'}(x)$ is invertible. Now, by the extreme value theorem, $P$ admits a minimal at some point $x\_0$ on the closed ball $\backslash overline\{B\}(0,\; r)$, which can be shown to lie in $B(0,\; r)$ using $2^\{-1\}|x|\; \backslash le\; |f(x)|$. Since $P\text{'}(x\_0)\; =\; 0$, $f(x\_0)\; =\; y$, which proves the claimed inclusion. $\backslash square$

Alternatively, one can deduce the theorem from the one over real numbers by Tarski's principle.{{citation needed|date=December 2024}}

• Nash–Moser theorem

{{reflist}}

• {{cite book |first=Carl B. |last=Allendoerfer |author-link=Carl B. Allendoerfer |title=Calculus of Several Variables and Differentiable Manifolds |location=New York |publisher=Macmillan |year=1974 |chapter=Theorems about Differentiable Functions |pages=54–88 |isbn=0-02-301840-2 }}
• {{cite book |first1=Peter |last1=Baxandall |author-link=Peter Baxandall |first2=Hans |last2=Liebeck |title=Vector Calculus |location=New York |publisher=Oxford University Press |year=1986 |chapter=The Inverse Function Theorem |isbn=0-19-859652-9 |pages=214–225 }}
• {{cite journal |last = Nijenhuis |first = Albert |author-link= Albert Nijenhuis |title = Strong derivatives and inverse mappings |journal = Amer. Math. Monthly |volume = 81 |year = 1974 |pages = 969–980 |doi = 10.2307/2319298 |issue = 9 |jstor = 2319298 |hdl = 10338.dmlcz/102482 |hdl-access = free }}
• {{citation

| last1 = Griffiths | first1 = Phillip

| last2 = Harris | first2 = Joseph

| isbn = 978-0-471-05059-9

| publisher = John Wiley & Sons

| title = Principles of Algebraic Geometry

| year = 1978}}.

• {{cite book |last1=Hirsch |first1=Morris W. |title=Differential Topology |date=1976 |publisher=Springer-Verlag |isbn=978-0-387-90148-0 |url=https://www.researchgate.net/publication/268035774 |language=en}}
• {{cite book |first1=Murray H. |last1=Protter |author-link=Murray H. Protter |first2=Charles B. Jr. |last2=Morrey |author-link2=Charles B. Morrey Jr. |title=Intermediate Calculus |location=New York |publisher=Springer |edition=Second |year=1985 |isbn=0-387-96058-9 |chapter=Transformations and Jacobians |pages=412–420 }}
• {{cite book |last1=Renardy |first1=Michael |last2=Rogers |first2=Robert C. |title = An Introduction to Partial Differential Equations | series = Texts in Applied Mathematics 13

| edition = Second |publisher = Springer-Verlag | location = New York |year = 2004 |pages = 337–338 |isbn = 0-387-00444-0 }}

• {{cite book |last = Rudin|first = Walter|author-link= Walter Rudin|title = Principles of mathematical analysis|url = https://archive.org/details/principlesofmath00rudi|url-access = registration|edition = Third |series = International Series in Pure and Applied Mathematics |publisher = McGraw-Hill Book | location = New York |year = 1976 |pages = [https://archive.org/details/principlesofmath00rudi/page/221 221]–223 | isbn=978-0-07-085613-4 }}
• {{cite book |title=Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus |last1=Spivak|first1=Michael|title-link=Calculus on Manifolds (book)|publisher= Benjamin Cummings |year=1965 |isbn=0-8053-9021-9 |location=San Francisco |author1-link=Michael Spivak }}

{{Functional analysis}}

{{Analysis in topological vector spaces}}

Category:Multivariable calculus

Category:Differential topology

Category:Inverse functions

Category:Theorems in real analysis

Category:Theorems in calculus

de:Satz von der impliziten Funktion#Satz von der Umkehrabbildung