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 The inverse function theorem is not often stated separately for one variable, because a stronger result is true: if a real-valued function of a single real variable has a derivative which is nonzero on an interval, then the function has an inverse throughout the interval. More precisely, suppose that is a real-valued differentiable function on an open interval , and is non-zero throughout . Then the image of the interval is another interval , is a bijection, and has a differentiable inverse function . 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 statement. The statement of the inverse function theorem is, roughly speaking, that if a real-valued function of a single real variable has a continuous derivative on an open interval , then it is locally invertible near each point where its derivative is non-zero. More precisely, around any point in where , there is a smaller interval on which the function is one-to-one and the image of the interval under is also an open interval . Thus maps the interval bijectively onto the interval . The inverse function theorem in this form follows at once from the stronger global result, by restricting the interval to a smaller interval around on which the derivative is non-zero throughout.
When is continuously differentiable, the inverse function is also continuously differentiable, and its derivative at any point in the interval is given by the inverse function rule:
The derivative being non-zero at a point is a sufficient condition for the function to have a local inverse, but it is not necessary. A function may be injective near a point while . An example is . But, for such a function, the inverse cannot be differentiable at , since if were differentiable at , then, by the chain rule, , which implies . (The situation is different for holomorphic functions; see Holomorphic inverse function theorem below.)
Several variables 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 (that is, the determinant of the Jacobian matrix of at is non-zero), then there exist open neighborhoods of in and of such that and is bijective.Theorem 1.1.7. in Writing , this means that the system of equations has a unique solution for in terms of when . Note that, unlike the case of a single variable, 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 in this formula is the existence and differentiability of . Assuming this, the inverse derivative formula follows from the chain rule applied to . (Indeed, ) 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 (i.e., the Jacobian matrix representing it has rank ) 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 (when is surjective), the point is called a regular value. Since , the first case is equivalent to saying is not in the image of critical points (a critical point is a point such that the kernel of 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 is surjective, we can find an (injective) 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 injective) 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 (and not invertible) on any interval containing . Intuitively, the slope 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 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 (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: 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 (a standard result in analysis). 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 (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,Theorem 17.7.2 in 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 part of the argument is the same as that in the previous proof). 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