Inverse function theorem


In real analysis, a branch of mathematics, the inverse function theorem is a theorem that asserts that, if a real function f has a continuous derivative near a point where its derivative is nonzero, then, near this point, f has an inverse function. The inverse function is also continuously differentiable, and the inverse function rule expresses its derivative as the multiplicative inverse of the derivative of f.
The theorem applies verbatim to complex-valued functions of a complex variable. It generalizes to functions from
n-tuples to n-tuples, and to functions between vector spaces of the same finite dimension, by replacing "derivative" with "Jacobian matrix" and "nonzero derivative" with "nonzero Jacobian determinant".
If the function of the theorem belongs to a higher differentiability class, the same is true for the inverse function. There are also versions of the inverse function theorem for holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces, and so forth.
The theorem was first established by Picard and Goursat using an iterative scheme: the basic idea is to prove a fixed point theorem using the contraction mapping theorem.

Statements

For functions of a single variable, the theorem states that if is a continuously differentiable function with nonzero derivative at the point ; then is injective in a neighborhood of, the inverse is continuously differentiable near, and the derivative of the inverse function at is the reciprocal of the derivative of at :
It can happen that a function may be injective near a point while. An example is. In fact, for such a function, the inverse cannot be differentiable at, since if were differentiable at, then, by the chain rule,, which implies.
For functions of more than one variable, the theorem states that if is a continuously differentiable function from an open subset of into, and the derivative is invertible at a point, then there exist neighborhoods of in and of such that and is bijective. Writing, this means that the system of equations has a unique solution for in terms of when. Note that the theorem does not say is bijective onto the image where is invertible but that it is locally bijective where is invertible.
Moreover, the theorem says that the inverse function is continuously differentiable, and its derivative at is the inverse map of ; i.e.,
In other words, if are the Jacobian matrices representing, this means:
The hard part of the theorem is the existence and differentiability of. Assuming this, the inverse derivative formula follows from the chain rule applied to. Since taking the inverse is infinitely differentiable, the formula for the derivative of the inverse shows that if is continuously times differentiable, with invertible derivative at the point, then the inverse is also continuously times differentiable. Here is a positive integer or.
There are two variants of the inverse function theorem. Given a continuously differentiable map, the first is
  • The derivative is surjective if and only if there exists a continuously differentiable function on a neighborhood of such that near,
and the second is
  • The derivative is injective if and only if there exists a continuously differentiable function on a neighborhood of such that near.
In the first case, the point is called a regular value. Since, the first case is equivalent to saying is not in the image of critical points . 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 is surjective, we can find an linear map such that. Define so that we have:
Thus, by the inverse function theorem, has inverse near ; i.e., near. The second case is seen in the similar way.

Example

Consider the vector-valued function defined by:
The Jacobian matrix of it at is:
with the determinant:
The determinant is nonzero everywhere. Thus the theorem guarantees that, for every point in, there exists a neighborhood about over which is invertible. This does not mean is invertible over its entire domain: in this case is not even injective since it is periodic:.

Counter-example

If one drops the assumption that the derivative is continuous, the function is no longer necessarily locally injective. For example and has discontinuous derivative
and which vanishes arbitrarily close to. These critical points are local max/min points of so is not one-to-one on any interval containing. Intuitively, the slope does not propagate to nearby points, where the slopes are governed by a weak but rapid oscillation.
If the derivative is continuous but zero at a point, the function is no longer necessarily locally injective. A real function that is locally constant at a point in the interior of its domain is not locally injective at but is trivially continuously differentiable at.

Methods of proof

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.
Since the fixed point theorem applies in infinite-dimensional settings, this proof generalizes immediately to the infinite-dimensional version of the inverse function theorem.
An alternate proof in finite dimensions hinges on the extreme value theorem for functions on a compact set. This approach has an advantage that the proof generalizes to a situation where there is no Cauchy completeness.
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.

Proof for single-variable functions

We want to prove the following: Let be an open set with a continuously differentiable function defined on, and suppose that. Then there exists an open interval with such that maps bijectively onto the open interval, and such that the inverse function is continuously differentiable, and for any, if is such that, then.
We may without loss of generality assume that. Given that is an open set and is continuous at, there exists such that and
In particular,
This shows that is strictly increasing for all. Let be such that. Then. By the intermediate value theorem, we find that maps the interval bijectively onto. Denote by and. Then is a bijection and the inverse exists. The fact that is differentiable follows from the differentiability of. In particular, the result follows from the fact that if is a strictly monotonic and continuous function that is differentiable at with, then is differentiable with, where . This completes the proof.

A proof using successive approximation

To prove existence, it can be assumed after an affine transformation that and, so that.
By the mean value theorem for vector-valued functions, for a differentiable function,. Setting, it follows that
Now choose so that for. Suppose that and define inductively by and. The assumptions show that if then
In particular implies. In the inductive scheme
and. Thus is a Cauchy sequence tending to. By construction as required.
To check that is C1, write so that
. By the inequalities above, so that.
On the other hand, if, then. Using the geometric series for, it follows that. But then
tends to 0 as and tend to 0, proving that is C1 with.
The proof above is presented for a finite-dimensional space, but applies equally well for Banach spaces. If an invertible function is Ck with, then so too is its inverse. This follows by induction using the fact that the map on operators is Ck for any .
The method of proof here can be found in the books of Henri Cartan, Jean Dieudonné, Serge Lang, Roger Godement and Lars Hörmander.

A proof using the contraction mapping principle

Here is a proof based on the contraction mapping theorem. Specifically, following T. Tao, it uses the following consequence of the contraction mapping theorem.
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 and. Let. The mean value inequality applied to says:
Since and is continuous, we can find an such that
for all in. Then the early lemma says that is injective on and. Then
is bijective and thus has an inverse. Next, we show the inverse is continuously differentiable. This time, let denote the inverse of and. For, we write or. Now, by the early estimate, we have
and so. Writing for the operator norm,
As, we have and is bounded. Hence, is differentiable at with the derivative. Also, is the same as the composition where ; so is continuous.
It remains to show the lemma. First, we have:
which is to say
This proves the first part. Next, we show. The idea is to note that this is equivalent to, given a point in, find a fixed point of the map
where such that and the bar means a closed ball. To find a fixed point, we use the contraction mapping theorem and checking that is a well-defined strict-contraction mapping is straightforward. Finally, we have: since
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.

Applications

Implicit function theorem

The inverse function theorem can be used to solve a system of equations
i.e., expressing as functions of, provided the Jacobian matrix is invertible. The implicit function theorem allows to solve a more general system of equations:
for in terms of. 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:
  • given a map, if, is continuously differentiable in a neighborhood of and the derivative of at is invertible, then there exists a differentiable map for some neighborhoods of such that. Moreover, if, then ; i.e., is a unique solution.
To see this, consider the map. By the inverse function theorem, has the inverse for some neighborhoods. We then have:
implying and Thus has the required property.

Giving a manifold structure

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. Indeed, let be such a smooth map from an open subset of . Fix a point in and then, by permuting the coordinates on, assume the matrix has rank. Then the map is such that has rank. Hence, by the inverse function theorem, we find the smooth inverse of defined in a neighborhood of. We then have
which implies
That is, after the change of coordinates by, is a coordinate projection. Moreover, since is bijective, the map
is bijective with the smooth inverse. That is to say, gives a local parametrization of around. Hence, is a manifold.
More generally, the theorem shows that if a smooth map is transversal to a submanifold, then the pre-image is a submanifold.

Global version

The inverse function theorem is a local result; it applies to each point. A priori, the theorem thus only shows the function is locally bijective. The next topological lemma can be used to upgrade local injectivity to injectivity that is global to some extent.
Proof: First assume is compact. If the conclusion of the theorem is false, we can find two sequences such that and each converge to some points in. Since is injective on,. Now, if is large enough, are in a neighborhood of where is injective; thus,, a contradiction.
In general, consider the set. It is disjoint from for any subset where is injective. Let be an increasing sequence of compact subsets with union and with contained in the interior of. Then, by the first part of the proof, for each, we can find a neighborhood of such that. Then has the required property.
The lemma implies the following global version of the inverse function theorem:
Note that if is a point, then the above is the usual inverse function theorem.

Holomorphic inverse function theorem

There is a version of the inverse function theorem for holomorphic maps.
The theorem follows from the usual inverse function theorem. Indeed, let denote the Jacobian matrix of in variables and for that in. Then we have, which is nonzero by assumption. Hence, by the usual inverse function theorem, is injective near with continuously differentiable inverse. By chain rule, with,
where the left-hand side and the first term on the right vanish since and are holomorphic. Thus, for each.
Similarly, there is the implicit function theorem for holomorphic functions.
As already noted earlier, it can happen that an injective smooth function has the inverse that is not smooth. This is not the case for holomorphic functions because of:

Formulations for manifolds

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, if the differential of,
is a linear isomorphism at a point in then there exists an open neighborhood of such that
is a diffeomorphism. Note that this implies that the connected components of and containing p and F have the same dimension, as is already directly implied from the assumption that dFp is an isomorphism.
If the derivative of is an isomorphism at all points in then the map is a local diffeomorphism.

Generalizations

Banach spaces

The inverse function theorem can also be generalized to differentiable maps between Banach spaces ' and '. Let ' be an open neighbourhood of the origin in ' and a continuously differentiable function, and assume that the Fréchet derivative of ' at 0 is a bounded linear isomorphism of ' onto '. Then there exists an open neighbourhood ' of in ' and a continuously differentiable map such that for all ' in '. Moreover, is the only sufficiently small solution ' of the equation.
There is also the inverse function theorem for Banach manifolds.

Constant rank theorem

The inverse 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. Specifically, if has constant rank near a point, then there are open neighborhoods of and of and there are diffeomorphisms and such that and such that the derivative is equal to. That is, "looks like" its derivative near. The set of points such that the rank is constant in a neighborhood of is an open dense subset of ; 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 is injective at a point, it is also injective in a neighborhood of, and hence the rank of is constant on that neighborhood, and the constant rank theorem applies.

Polynomial functions

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, 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.

Selections

When with, is times continuously differentiable, and the Jacobian at a point is of rank, the inverse of may not be unique. However, there exists a local selection function such that for all in a neighborhood of,, is times continuously differentiable in this neighborhood, and .

Over a real closed field

The inverse function theorem also holds over a real closed field k. Precisely, the theorem holds for a semialgebraic map between open subsets of 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, the Cauchy completeness is used only to establish the inclusion. Here, we shall directly show instead. Given a point in, consider the function defined on a neighborhood of. If, then and so, since is invertible. Now, by the extreme value theorem, admits a minimal at some point on the closed ball, which can be shown to lie in using. Since,, which proves the claimed inclusion.
Alternatively, one can deduce the theorem from the one over real numbers by Tarski's principle.