In mathematical analysis, the inverse function theorem gives sufficient conditions for a function to have an inverse function. The essential idea is that if the best linear approximation to the function at a point is invertible, then with sufficient regularity assumptions, the function should also be invertible near that point. In its simplest form, the theorem states that if a real function f is differentiable in an open interval, with a continuous derivative, then in a neighborhood of any point where the derivative is not zero, 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 (of real or complex numbers) 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

One variable

is a bijection. This theorem is true because the non-vanishing of the derivative implies that it must be entirely of one sign (positive or negative) according to Darboux's theorem, and therefore the function must be strictly monotone, and thus one-to-one.

The inverse function theorem is a weaker local, then it is locally is one-to-one bijectively on which the derivative is non-zero throughout.

is given by the inverse function rule

. (The situation is different for holomorphic functions; see Holomorphic inverse function theorem below.)

Several variables

, and the derivative is invertible at a point (that is, the determinant of the Jacobian matrix of at is bijective., this means that the system of . Note that, unlike the case of a single variable, the theorem does not is invertible.

; i.e.,

, this means:

. Assuming this, the inverse derivative formula follows from the chain rule times differentiable, with invertible derivative at the point .

There are two variants of the inverse function theorem., the first is

• ,

and the second is

• .

is called a regular value is not in the image of critical points is nonzero). The statement in the first case is a special case of the submersion theorem.

so that we have:

is injective) is seen in the similar way.

Example

Consider the vector-valued function defined by:

is:

with the determinant:

is nonzero everywhere. Thus the theorem guarantees that, for every point , 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.

Counter-example

does not propagate to nearby points, where the slopes rapidly oscillate between -1 and 3 (approximately).

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.

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 (which can also be used as the key step in the proof of existence and uniqueness of solutions to ordinary differential equations).

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 (see Generalizations below).

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 (see ).

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: .

(a standard result in analysis). This completes the proof.

A proof using successive approximation

.

By the mean value theorem for vector-valued functions, it follows that

then

1. .

is a Cauchy sequence as required.

is C1. Using the geometric series. But then

is C1.

The proof above is presented for a finite-dimensional space, but applies equally well for Banach spaces is Ck on operators is Ck (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). 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.

. The mean value inequality says:

such that

. Then

. Now, by the early estimate, we have

for the operator norm,

is continuous.

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

which is to say

, find a fixed point of the map

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

, provided the Jacobian matrix is invertible. The implicit function theorem allows to solve a more general system of equations:

. 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:

• is a unique solution.

. We then have:

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.. We then have

which implies

is a coordinate projection (this fact is known as the submersion theorem is bijective, the map

(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.)

is a submanifold.

Global version

The inverse function theorem is a local result; it applies to each point. A priori 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.

Proof: is compact, a contradiction.

(See also for an alternative approach.)

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

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.

,

Similarly, there is the implicit function theorem for holomorphic functions.

in a real variable). 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), if the differential,

is a linear isomorphism such that

is a diffeomorphism. Note that this implies that the connected components of and containing p and F(p) 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.

Banach spaces

The inverse function theorem can be generalized to differentiable maps between Banach spaces and .

Let be an open neighbourhood of the origin in of at 0 is a bounded linear isomorphism of onto . Then there exists an open neighbourhood in for all in is the only sufficiently small solution .

There is also the inverse function theorem for Banach manifolds.

Example

Then is continuously Fréchet differentiable, the derivative is the operator . This is an isomorphism from to Therefore the inverse function theorem for Banach spaces implies that is locally invertible near . In particular, for every sufficiently small has a unique small solution , and this solution depends continuously differentiably on .

Generalizations

Constant rank theorem

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.; this is a consequence of semicontinuity of the rank function. Thus the constant rank theorem applies to a generic point of the domain.

is constant on that neighborhood, and the constant rank theorem applies.

Polynomial functions

If it were 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. In July 2026, a counterexample has been provided that shows that the conjecture is false for three variables or more. For two variables, it remains unknown whether the conjecture is true or false. For a single variable, the conjecture is trivially true.

Selections

times continuously differentiable is of rank may not be unique. However, there exists a local selection function in a neighborhood is the Moore–Penrose pseudoinverse).

Over a real closed field

The inverse function theorem also holds over a real closed field k (or an o-minimal structure). 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

Alternatively, one can deduce the theorem from the one over real numbers by Tarski's principle.

See also

• Nash–Moser theorem

Notes

References

• .

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