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The function that associates with each of the elements of the set of values of a given function the set of all elements from the domain of definition of the given function that are mapped onto it, that is, its complete inverse image. If the given function is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i0523601.png" />, then the inverse function is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i0523602.png" />. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i0523603.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i0523604.png" /> is the range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i0523605.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i0523606.png" />, then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i0523607.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i0523608.png" />.
+
==Definition==
 +
A map $f:X \to Y$ is called invertible if for every $y\in Y$ there exists one and only one $x\in X$ such that $f(x) = y$. When the map is invertible the ''inverse function'' is the map $g:Y \to X$ which maps any $x$ precisely into the element $y$ such that $f(x) =y$. The inverse function $g$ is then characterized by the following two properties:
 +
\[
 +
\begin{array}{ll}
 +
g(f(x))=x\qquad &\forall x\in X\\
 +
f(g(y))=y\qquad &\forall y\in Y\, .
 +
\end{array}
 +
\]
 +
The map $g$ is then usually denoted by $f^{-1}$ and hence it is commonly written $f^{-1} (y) = x$. The inverse function must not be confused with the ''preimage'' of $y$, which is usually denoted by $f^{-1} (\{y\})$ (although many authors still use the notation $f^{-1} (y)$), and it consists of the subset of $X$ defined by
 +
\[
 +
f^{-1} (\{y\}) := \{x: f(x) = y\}\, .
 +
\]
 +
An invertible map is by definition a map such that the preimage of each element has cardinality $1$. Although rarely in use nowadays, when the map $f$ is [[Surjection|surjective]] the set-valued map which to each $y$ associates the preimage of $y$ is sometimes also called inverse.
  
If for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i0523609.png" /> the complete inverse image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236010.png" /> consists of precisely one element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236011.png" />, that is, if the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236012.png" /> is a bijection, then the inverse function is single-valued, otherwise it is many-valued.
+
Often, if a map $f:X\to Y$ is injective, then we can consider it as an invertible map from $X$ into its range $f(X)$
 +
and its inverse, defined on $f(X)$, is also denoted by $f^{-1}$.
  
If the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236014.png" /> are subsets of the real line (or, more generally, of some ordered sets) then strict monotonicity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236015.png" /> is a necessary and sufficient condition for the existence of an inverse function that is also strictly monotone.
+
===Injectivity and surjectivity, left and right inverse===
 +
When a map $f$ is ''onto'', namely for every $y\in Y$ there exists at least one $x$ such that $f(x)=y$, then $f$ is called [[Surjection|surjective]]. Surjectivity is characterized by the property that the preimage of any element is nonempty. It is also characterized by the existence of a ''right inverse'', namely a map $g: Y \to X$ such that $f(g(y)) = y$ for every $y\in Y$ (this is a consequence of the [[Axiom of choice]]).  
  
A number of properties of the inverse function can be determined from the corresponding properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236016.png" />. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236017.png" /> is strictly monotone and continuous on some interval of the real line, then its inverse is also monotone and continuous on the corresponding interval. If a one-to-one mapping of a compactum onto a Hausdorff topological space is continuous, then the inverse mapping is also continuous. That is, the original mapping is a [[Homeomorphism|homeomorphism]] onto its image. When the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236018.png" /> is a one-to-one bounded linear operator mapping a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236019.png" /> onto a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236020.png" />, then the inverse operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236021.png" /> is also linear and bounded.
+
When a map $f$ is ''one-to-one'', namely $f(x)\neq f(y) \rightarrow x\neq y$, then $f$ is called [[Injection|injective]]. Injectivity is characterized by the property that the preimage of any element has never cardinality larger than $1$. It is also characterized by the existence of a ''left inverse'', namely a function $g: Y\to X$ such that $g(f(x)) =x$ for every $x\in X$.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236022.png" /> be a continuous mapping of the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236023.png" /> of a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236025.png" />, with a sufficiently smooth boundary in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236026.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236027.png" /> be differentiable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236028.png" /> and map the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236029.png" /> onto the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236030.png" /> and suppose that the set of zeros of the Jacobian of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236031.png" /> is an isolated set; then if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236032.png" /> is one-to-one on the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236033.png" />, it is one-to-one on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236034.png" />. For the existence of a local inverse mapping in a neighbourhood of a given point it is sufficient that the Jacobian of the mapping does not vanish in some neighbourhood of this point. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236036.png" />, is a differentiable mapping with non-zero Jacobian at all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236037.png" />, then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236038.png" /> there exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236039.png" /> such that the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236040.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236041.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236042.png" /> is a one-to-one mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236043.png" /> onto some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236044.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236045.png" />, and the inverse mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236046.png" /> is also differentiable (on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236047.png" />). This theorem can be generalized to the infinite-dimensional case: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236049.png" /> be complete normed spaces, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236050.png" /> be an open set and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236051.png" /> be a continuously-differentiable mapping. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236052.png" /> is an invertible element in the space of bounded linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236053.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236054.png" /> is the [[Fréchet derivative|Fréchet derivative]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236055.png" />, then there exists neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236057.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236059.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236061.png" /> respectively, such that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236062.png" /> and its [[Inverse mapping|inverse mapping]] are continuously-differentiable homeomorphisms.
+
It follows therefore that a map is invertible if and only if it is injective and surjective at the same time. An invertible map is also called [[Bijection|bijective]].  
  
====References====
+
===Behavior under composition===
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,   S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR></table>
+
The composition of two surjective maps is also surjective. Similarly the composition of two injective maps is also injective. Consequently, the composition of two invertible maps is also invertible. Moreover, let $f:X\to Y$ and $g:Y\to Z$ be two invertible maps. The inverse of $g\circ f$ is then given by $f^{-1} \circ g^{-1}$.
  
 +
===Group structure of invertible maps===
 +
Let $X$ be a set and consider the set $\mathcal{I} (X)$ of maps $f:X\to X$ which are invertible. $\mathcal{I} (X)$ with the operation of composition is then a [[Group|group]]: the identity is the map $x\mapsto x$ and the inverse of an element $f\in \mathcal{I} (X)$ is precisely the inverse function.
  
 +
===Involution===
 +
An [[Involution|involution]] is map $f:X \to X$ which is the inverse of itself, namely such that $f(f(x))=x$ for every $x\in X$. If ${\rm id}$ denotes the identity map, then the property of being an involution can be expressed with the formula $f\circ f = {\rm id}$. A notable example of involution is the conjugation on the [[Complex number|complex plane]]:
 +
\[
 +
\mathbb C \ni z = x+i y \quad\mapsto\quad \bar{z} = x - i y\, .
 +
\]
  
====Comments====
+
==Real-valued functions of one real variable==
The assertions in the last paragraph of the main article are known under the (collective) name of inverse-function theorem.
+
Let $I$ and $J$ be two intervals and $f: I \to J$ a continuous surjective function. Then $f$ is injective if and only if it is strictly [[Monotone function|monotone]] and in that case the inverse function $f^{-1}$ is also continuous.  
  
Nowadays the term  "function"  is usually reserved for those relations that are single-valued, and the term  "mapping"  is one of its synonyms. When this is done, only bijections (one-to-one onto functions) have inverses that are functions. In all other cases, the inverse relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052360/i05236063.png" /> (called a many-valued function in the main article) is not a function unless, as is sometimes done, it is regarded as being set-valued. Then arises the important but simple distinction between a singleton set and its unique element.
+
===Differentiable functions===
 +
It follows from [[Finite-increments formula|Lagrange theorem]] that if $f: I \to \mathbb R$ is differentiable and $f'(y) \neq 0$ for every $y$, then $f$ is necessarily strictly monotone. Hence, if we set $J:= f(I)$, then its inverse $f^{-1} : J \to I$ is also differentiable. An important generalization of this fact to functions of several variables is the Inverse function theorem, Theorem 2 below.
 +
 
 +
===Formula for the derivative of the inverse===
 +
Under the assumptions above we have the formula
 +
\begin{equation}\label{e:derivative_inverse}
 +
(f^{-1})' (y) = \frac{1}{f'(f^{-1}(y))}
 +
\end{equation}
 +
for the derivative of the inverse.
 +
 
 +
In fact, the chain rule guarantees that, whenever $f$ is invertible and both $f$ and $f^{-1}$ are differentiable, then both $f'$ and $(f^{-1})'$ are everywhere nonvanishing.
 +
 
 +
===Higher differentiability===
 +
Under the assumptions above, the map $f^{-1}$ inherits the differentiability properties of $f$. The higher oder derivatives of $f^{-1}$ can be computed in terms of the derivatives of $f$, as can be seen by differentiating further the formula \eqref{e:derivative_inverse}.
 +
 
 +
==Homeomorphisms==
 +
If $X$ and $Y$ are topological spaces and $f:X\to Y$ is a continuous invertible map with a continuous inverse, then $f$ is called an [[Homeomorphism|homeomorphism]].
 +
 
 +
===A criterion for the continuity of the inverse===
 +
A very useful theorem in general topology is the following
 +
 
 +
'''Theorem 1'''
 +
Assume $X$ is a [[Compact space|compact topological space]] and $Y$ a [[Hausdorff space]]. Then, any continuous invertible map $f:X\to Y$ is necessarily an homeomorphism.
 +
 
 +
==Diffeomorphisms==
 +
Let $U, V \subset \mathbb R^n$ be open and let $f: U \to V$ be a continuously differentiable map with a continuously differentiable inverse. Then $f$ is called a [[Diffeomorphism|diffeomorphism]] of $U$ onto $V$. This concept is usually extended to maps between general [[Differentiable manifold|manifolds]] using charts.
 +
 
 +
Observe that if $f$ is a diffeomorphism, then the chain rule implies necessarily that the differentiable of $f$ is invertible at every point (namely its [[Jacobian|Jacobi matrix]] is invertible). Moreover, it gives the formula
 +
\[
 +
\left. d (f^{-1})\right|_y = \left[\left. df\right|_{f^{-1} (y)}\right]^{-1}\, .
 +
\]
 +
A partial converse of this statement is given by the important Implicit function theorem.
 +
 
 +
===Higher differentiability properties===
 +
If $f$ is a diffeomorphism, then $f^{-1}$ inherits the differentiability properties of $f$. Namely, if $f$ is $N$ times differentiable, then so is $f^{-1}$, and if $f$ is analytic, so is $f^{-1}$.
 +
 
 +
===Inverse function theorem===
 +
A very useful fact in analysis is that $C^1$ maps $f$ such that $\left. df\right|_{x_0}$ is invertible at some point $x_0$ are ''local diffeomorphisms''. More precisely
 +
 
 +
'''Theorem 2'''
 +
Let $U\subset \mathbb R^n$ be an open set and $f: U \to \mathbb R^n$ be a $C^1$ map such that $df|_{x_0}$ is invertible at some point $x_0\in U$. Then there is a neighborhood $V$ of $x_0$ such that $f|_V$ is a diffeomorphism of $V$ onto $f(V)$.
 +
 
 +
The latter theorem is usually linked to the [[Implicit function|Implicit function theorem]].
 +
 
 +
Unlike in the $1$-dimensional case, the condition that the differential is invertible at ''every point'' does not guarantee the ''global invertibility'' of the map. Indeed, a famous example is the exponential map on the complex plane:
 +
\[
 +
{\rm exp}: \mathbb C \in z \mapsto e^z \in \mathbb C\, .
 +
\]
 +
This map can be considered as a map from $\mathbb R^2$ onto $\mathbb R^2\setminus \{0\}$. The explicit formula is then given by
 +
\[
 +
(x,y)\mapsto (e^x \cos y, e^x\sin y)\, .
 +
\]
 +
It can be readily checked that the differential is invertible at every point. On the other hand the map is not injective (in fact each $(x,y)\in \mathbb R^2\setminus \{0\}$ has infinitely many counterimages).
 +
 
 +
====A criterion for global invertibility====
 +
A useful criterion for global invertibility is the following.
 +
 
 +
'''Theorem 3'''
 +
Let $U, V\subset \mathbb R^n$ be two bounded open sets with $C^1$ boundaries and $f: \overline{U}\to \overline{V}$ a continuous map such that
 +
* $f$ is $C^1$ in $U$;
 +
* $f (\partial U) = \partial V$ and $f$ is injective on $\partial U$;
 +
* The differential of $f$ is invertible at any $x\in U$ except for a finite set of points.
 +
Then $f$ is injective.
 +
 
 +
==The inverse function theorem in infinite dimension==
 +
The implicit function theorem has been successfully generalized in a variety of infinite-dimensional situations, which proved to be extremely useful in modern mathematics. The most straightforward generalization is the following
 +
(cf. [[Implicit function]]):
 +
 
 +
'''Theorem 4'''
 +
Let $X$ and $Y$ be two [[Banach space|Banach spaces]], $U\subset X$ an open subset and $f:U\to Y$ a continuously differentiable map in the [[Frechet differential|sense of Frechet]]. If the Frechet derivative of $f$ at some point $x_0$ has an inverse which is a [[Bounded operator|bounded linear operator]], then there is a neighborhood $V$ of $x_0$ such that $f|_V$ is invertible on $f(V)$.
 +
 
 +
This is often called ''soft inverse function theorem'', since it can be proved using essentially the same techniques as those in the finite-dimensional version. A much more difficult generalization (to "tame" [[Frechet space|Frechet spaces]]) is given by the ''hard inverse function theorems'', which followed a pioneering idea of Nash in {{Cite|Na}} and was extended further my Moser, see [[Nash-Moser iteration]].
 +
 
 +
==References==
 +
{|
 +
|-
 +
|valign="top"|{{Ref|Ke}}|| J.L. Kelley,  "General topology" , v. Nostrand  (1955)
 +
|-
 +
|valign="top"|{{Ref|KF}}|| A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions  and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)
 +
|-
 +
|valign="top"|{{Ref|Mo}}|| J. Moser, "A rapidly convergent iteration method and non-linear partial differential equations. I", ''Ann. Scuola Norm. Sup. Pisa (3)'' '''20''' (1966) pp. 265-315, {{MR|0199523}}
 +
|-
 +
|valign="top"|{{Ref|Mo1}}|| J. Moser, "A rapidly convergent iteration method and non-linear partial  differential equations. II", ''Ann. Scuola Norm. Sup. Pisa (3)'' '''20'''  (1966) pp. 499-535, {{MR|0206461}}
 +
|-
 +
|valign="top"|{{Ref|Na}}|| J, Nash, "The imbedding problem for Riemannian manifolds", ''Ann. of Math.'' '''63''' (1956) pp. 20-63, {{MR|0075639}}
 +
|-
 +
|valign="top"|{{Ref|Ru}}|| W. Rudin,  "Functional analysis" , McGraw-Hill  (1973)
 +
|-
 +
|valign="top"|{{Ref|Ru1}}|| W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 75–78 {{MR|0385023}}
 +
|-
 +
|}

Revision as of 13:56, 9 December 2013

2020 Mathematics Subject Classification: Primary: 03Exx [MSN][ZBL] 2020 Mathematics Subject Classification: Primary: 54C05 Secondary: 26B10 [MSN][ZBL]

Definition

A map $f:X \to Y$ is called invertible if for every $y\in Y$ there exists one and only one $x\in X$ such that $f(x) = y$. When the map is invertible the inverse function is the map $g:Y \to X$ which maps any $x$ precisely into the element $y$ such that $f(x) =y$. The inverse function $g$ is then characterized by the following two properties: \[ \begin{array}{ll} g(f(x))=x\qquad &\forall x\in X\\ f(g(y))=y\qquad &\forall y\in Y\, . \end{array} \] The map $g$ is then usually denoted by $f^{-1}$ and hence it is commonly written $f^{-1} (y) = x$. The inverse function must not be confused with the preimage of $y$, which is usually denoted by $f^{-1} (\{y\})$ (although many authors still use the notation $f^{-1} (y)$), and it consists of the subset of $X$ defined by \[ f^{-1} (\{y\}) := \{x: f(x) = y\}\, . \] An invertible map is by definition a map such that the preimage of each element has cardinality $1$. Although rarely in use nowadays, when the map $f$ is surjective the set-valued map which to each $y$ associates the preimage of $y$ is sometimes also called inverse.

Often, if a map $f:X\to Y$ is injective, then we can consider it as an invertible map from $X$ into its range $f(X)$ and its inverse, defined on $f(X)$, is also denoted by $f^{-1}$.

Injectivity and surjectivity, left and right inverse

When a map $f$ is onto, namely for every $y\in Y$ there exists at least one $x$ such that $f(x)=y$, then $f$ is called surjective. Surjectivity is characterized by the property that the preimage of any element is nonempty. It is also characterized by the existence of a right inverse, namely a map $g: Y \to X$ such that $f(g(y)) = y$ for every $y\in Y$ (this is a consequence of the Axiom of choice).

When a map $f$ is one-to-one, namely $f(x)\neq f(y) \rightarrow x\neq y$, then $f$ is called injective. Injectivity is characterized by the property that the preimage of any element has never cardinality larger than $1$. It is also characterized by the existence of a left inverse, namely a function $g: Y\to X$ such that $g(f(x)) =x$ for every $x\in X$.

It follows therefore that a map is invertible if and only if it is injective and surjective at the same time. An invertible map is also called bijective.

Behavior under composition

The composition of two surjective maps is also surjective. Similarly the composition of two injective maps is also injective. Consequently, the composition of two invertible maps is also invertible. Moreover, let $f:X\to Y$ and $g:Y\to Z$ be two invertible maps. The inverse of $g\circ f$ is then given by $f^{-1} \circ g^{-1}$.

Group structure of invertible maps

Let $X$ be a set and consider the set $\mathcal{I} (X)$ of maps $f:X\to X$ which are invertible. $\mathcal{I} (X)$ with the operation of composition is then a group: the identity is the map $x\mapsto x$ and the inverse of an element $f\in \mathcal{I} (X)$ is precisely the inverse function.

Involution

An involution is map $f:X \to X$ which is the inverse of itself, namely such that $f(f(x))=x$ for every $x\in X$. If ${\rm id}$ denotes the identity map, then the property of being an involution can be expressed with the formula $f\circ f = {\rm id}$. A notable example of involution is the conjugation on the complex plane: \[ \mathbb C \ni z = x+i y \quad\mapsto\quad \bar{z} = x - i y\, . \]

Real-valued functions of one real variable

Let $I$ and $J$ be two intervals and $f: I \to J$ a continuous surjective function. Then $f$ is injective if and only if it is strictly monotone and in that case the inverse function $f^{-1}$ is also continuous.

Differentiable functions

It follows from Lagrange theorem that if $f: I \to \mathbb R$ is differentiable and $f'(y) \neq 0$ for every $y$, then $f$ is necessarily strictly monotone. Hence, if we set $J:= f(I)$, then its inverse $f^{-1} : J \to I$ is also differentiable. An important generalization of this fact to functions of several variables is the Inverse function theorem, Theorem 2 below.

Formula for the derivative of the inverse

Under the assumptions above we have the formula \begin{equation}\label{e:derivative_inverse} (f^{-1})' (y) = \frac{1}{f'(f^{-1}(y))} \end{equation} for the derivative of the inverse.

In fact, the chain rule guarantees that, whenever $f$ is invertible and both $f$ and $f^{-1}$ are differentiable, then both $f'$ and $(f^{-1})'$ are everywhere nonvanishing.

Higher differentiability

Under the assumptions above, the map $f^{-1}$ inherits the differentiability properties of $f$. The higher oder derivatives of $f^{-1}$ can be computed in terms of the derivatives of $f$, as can be seen by differentiating further the formula \eqref{e:derivative_inverse}.

Homeomorphisms

If $X$ and $Y$ are topological spaces and $f:X\to Y$ is a continuous invertible map with a continuous inverse, then $f$ is called an homeomorphism.

A criterion for the continuity of the inverse

A very useful theorem in general topology is the following

Theorem 1 Assume $X$ is a compact topological space and $Y$ a Hausdorff space. Then, any continuous invertible map $f:X\to Y$ is necessarily an homeomorphism.

Diffeomorphisms

Let $U, V \subset \mathbb R^n$ be open and let $f: U \to V$ be a continuously differentiable map with a continuously differentiable inverse. Then $f$ is called a diffeomorphism of $U$ onto $V$. This concept is usually extended to maps between general manifolds using charts.

Observe that if $f$ is a diffeomorphism, then the chain rule implies necessarily that the differentiable of $f$ is invertible at every point (namely its Jacobi matrix is invertible). Moreover, it gives the formula \[ \left. d (f^{-1})\right|_y = \left[\left. df\right|_{f^{-1} (y)}\right]^{-1}\, . \] A partial converse of this statement is given by the important Implicit function theorem.

Higher differentiability properties

If $f$ is a diffeomorphism, then $f^{-1}$ inherits the differentiability properties of $f$. Namely, if $f$ is $N$ times differentiable, then so is $f^{-1}$, and if $f$ is analytic, so is $f^{-1}$.

Inverse function theorem

A very useful fact in analysis is that $C^1$ maps $f$ such that $\left. df\right|_{x_0}$ is invertible at some point $x_0$ are local diffeomorphisms. More precisely

Theorem 2 Let $U\subset \mathbb R^n$ be an open set and $f: U \to \mathbb R^n$ be a $C^1$ map such that $df|_{x_0}$ is invertible at some point $x_0\in U$. Then there is a neighborhood $V$ of $x_0$ such that $f|_V$ is a diffeomorphism of $V$ onto $f(V)$.

The latter theorem is usually linked to the Implicit function theorem.

Unlike in the $1$-dimensional case, the condition that the differential is invertible at every point does not guarantee the global invertibility of the map. Indeed, a famous example is the exponential map on the complex plane: \[ {\rm exp}: \mathbb C \in z \mapsto e^z \in \mathbb C\, . \] This map can be considered as a map from $\mathbb R^2$ onto $\mathbb R^2\setminus \{0\}$. The explicit formula is then given by \[ (x,y)\mapsto (e^x \cos y, e^x\sin y)\, . \] It can be readily checked that the differential is invertible at every point. On the other hand the map is not injective (in fact each $(x,y)\in \mathbb R^2\setminus \{0\}$ has infinitely many counterimages).

A criterion for global invertibility

A useful criterion for global invertibility is the following.

Theorem 3 Let $U, V\subset \mathbb R^n$ be two bounded open sets with $C^1$ boundaries and $f: \overline{U}\to \overline{V}$ a continuous map such that

  • $f$ is $C^1$ in $U$;
  • $f (\partial U) = \partial V$ and $f$ is injective on $\partial U$;
  • The differential of $f$ is invertible at any $x\in U$ except for a finite set of points.

Then $f$ is injective.

The inverse function theorem in infinite dimension

The implicit function theorem has been successfully generalized in a variety of infinite-dimensional situations, which proved to be extremely useful in modern mathematics. The most straightforward generalization is the following (cf. Implicit function):

Theorem 4 Let $X$ and $Y$ be two Banach spaces, $U\subset X$ an open subset and $f:U\to Y$ a continuously differentiable map in the sense of Frechet. If the Frechet derivative of $f$ at some point $x_0$ has an inverse which is a bounded linear operator, then there is a neighborhood $V$ of $x_0$ such that $f|_V$ is invertible on $f(V)$.

This is often called soft inverse function theorem, since it can be proved using essentially the same techniques as those in the finite-dimensional version. A much more difficult generalization (to "tame" Frechet spaces) is given by the hard inverse function theorems, which followed a pioneering idea of Nash in [Na] and was extended further my Moser, see Nash-Moser iteration.

References

[Ke] J.L. Kelley, "General topology" , v. Nostrand (1955)
[KF] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
[Mo] J. Moser, "A rapidly convergent iteration method and non-linear partial differential equations. I", Ann. Scuola Norm. Sup. Pisa (3) 20 (1966) pp. 265-315, MR0199523
[Mo1] J. Moser, "A rapidly convergent iteration method and non-linear partial differential equations. II", Ann. Scuola Norm. Sup. Pisa (3) 20 (1966) pp. 499-535, MR0206461
[Na] J, Nash, "The imbedding problem for Riemannian manifolds", Ann. of Math. 63 (1956) pp. 20-63, MR0075639
[Ru] W. Rudin, "Functional analysis" , McGraw-Hill (1973)
[Ru1] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 75–78 MR0385023
How to Cite This Entry:
Inverse function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inverse_function&oldid=30878
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article