Inverse function

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 $y$ precisely into the element $x$ such that $f(x) =y$. The inverse function $g$ is then characterized by the following property: \[ \begin{array}{ll} g(f(x))=x\qquad &\forall x\in X\\ f(g(y))=y\qquad &\forall y\in Y\, . \end{array} \] 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 a 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 differential 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

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[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
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