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 : An alternate version, which assumes that is continuous and injective near, and differentiable at with a non-zero derivative, will also result in being invertible near, with an inverse that's similarly continuous and injective, and where the above formula would apply as well. As a corollary, we see clearly that if is -th differentiable, with nonzero derivative at the point, then is invertible in a neighborhood of, the inverse is also -th differentiable. Here is a positive integer or. 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 , then is invertible near : an inverse function to is defined on some neighborhood of. Writing, this means that the system of equations has a unique solution for in terms of, provided that 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 :
Example
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:.
Counter-example
If one drops the assumption that the derivative is continuous, the function no longer need be invertible. 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.
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. 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.
Generalizations
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.
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.
Banach manifolds
These two directions of generalization can be combined in the inverse function theorem for Banach manifolds.
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. 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 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.
Holomorphic functions
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.
