Inverse function theorem
In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point. The theorem also gives a formula for the derivative of the inverse function. In multivariable calculus, this theorem can be generalized to any continuously differentiable, vector-valued function whose Jacobian determinant is nonzero at a point in its domain, giving a formula for the Jacobian matrix of the inverse. There are also versions of the inverse function theorem for complex holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces, and so forth.
Statement of the theorem
For functions of a single variable, the theorem states that if is a continuously differentiable function with nonzero derivative at the point , then is invertible in a neighborhood of , the inverse is continuously differentiable, and the derivative of the inverse function at is the reciprocal of the derivative of at :
For functions of more than one variable, the theorem states that if is a continuously differentiable function from an open set of into , and the total derivative is invertible at a point (i.e., the Jacobian determinant of at is non-zero), then is invertible near : an inverse function to is defined on some neighborhood of . Writing , this means the system of n equations has a unique solution for in terms of , provided we restrict and to small enough neighborhoods of and , respectively. In the infinite dimensional case, the theorem requires the extra hypothesis that the Fréchet derivative of at has a bounded inverse.
Finally, the theorem says that the inverse function is continuously differentiable, and its Jacobian derivative at is the matrix inverse of the Jacobian of at :
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 :
Consider the vector-valued function defined by:
The Jacobian matrix is:
with Jacobian 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: .
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. 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.
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 (of class ), if the differential of ,
is a linear isomorphism at a point in then there exists an open neighborhood of such that
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 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 .
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. 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 . Semicontinuity of the rank function implies that there is an open dense subset of the domain of on which the derivative has constant rank. Thus the constant rank theorem applies to a generic point of the domain.
When the derivative of is injective (resp. surjective) at a point , it is also injective (resp. surjective) in a neighborhood of , and hence the rank of is constant on that neighborhood, and the constant rank theorem applies.
If a holomorphic function is defined from an open set of into , and the Jacobian matrix of complex derivatives is invertible at a point , then is an invertible function near . This follows immediately from the real multivariable version of the theorem. One can also show that the inverse function is again holomorphic.
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