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.
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.
is invertible at a point a (that is, the determinant of the Jacobian matrix of f at a is non-zero), then there exist neighborhoods
Assuming this, the inverse derivative formula follows from the chain rule applied to
times differentiable, with invertible derivative at the point a, then the inverse is also continuously
This does not mean F is invertible over its entire domain: in this case F is not even injective since it is periodic:
If one drops the assumption that the derivative is continuous, the function no longer need be invertible.
does not propagate to nearby points, where the slopes are governed by a weak but rapid oscillation.
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).
[2][3] 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[4] (see Generalizations below).
An alternate proof in finite dimensions hinges on the extreme value theorem for functions on a compact set.
[5] This approach has an advantage that the proof generalizes to a situation where there is no Cauchy completeness (see § Over a real closed field).
[1][7] The method of proof here can be found in the books of Henri Cartan, Jean Dieudonné, Serge Lang, Roger Godement and Lars Hörmander.
To find a fixed point, we use the contraction mapping theorem and checking that
The inverse function theorem can be used to solve a system of equations i.e., expressing
The implicit function theorem allows to solve a more general system of equations: for
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.
[11] The inverse function theorem is a local result; it applies to each point.
(or, more generally, a topological space admitting an exhaustion by compact subsets) and
The inverse function theorem can also be generalized to differentiable maps between Banach spaces X and Y.
is an open dense subset of M; this is a consequence of semicontinuity of the rank function.
Thus the constant rank theorem applies to a generic point of the domain.
If it is true, the Jacobian conjecture would be a variant of the inverse function theorem for polynomials.
[23] The inverse function theorem also holds over a real closed field k (or an O-minimal structure).
[24] Precisely, the theorem holds for a semialgebraic (or definable) map between open subsets of
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 § A proof using the contraction mapping principle, the Cauchy completeness is used only to establish the inclusion
Alternatively, one can deduce the theorem from the one over real numbers by Tarski's principle.
