Inverse function

The function that associates with each of the elements of the set of values of a given function the set of all elements from the domain of definition of the given function that are mapped onto it, that is, its complete inverse image. If the given function is denoted by , then the inverse function is denoted by . Thus, if and is the range of , , then for any one has .

If for all the complete inverse image of consists of precisely one element , that is, if the mapping is a bijection, then the inverse function is single-valued, otherwise it is many-valued.

If the sets and are subsets of the real line (or, more generally, of some ordered sets) then strict monotonicity of is a necessary and sufficient condition for the existence of an inverse function that is also strictly monotone.

A number of properties of the inverse function can be determined from the corresponding properties of . For example, if is strictly monotone and continuous on some interval of the real line, then its inverse is also monotone and continuous on the corresponding interval. If a one-to-one mapping of a compactum onto a Hausdorff topological space is continuous, then the inverse mapping is also continuous. That is, the original mapping is a homeomorphism onto its image. When the mapping is a one-to-one bounded linear operator mapping a Banach space onto a Banach space , then the inverse operator is also linear and bounded.

Let be a continuous mapping of the closure of a bounded domain , , with a sufficiently smooth boundary in , let be differentiable in and map the boundary of onto the boundary of and suppose that the set of zeros of the Jacobian of is an isolated set; then if is one-to-one on the boundary of , it is one-to-one on . For the existence of a local inverse mapping in a neighbourhood of a given point it is sufficient that the Jacobian of the mapping does not vanish in some neighbourhood of this point. If , , is a differentiable mapping with non-zero Jacobian at all points , then for any there exists a neighbourhood such that the restriction of to is a one-to-one mapping of onto some neighbourhood of , and the inverse mapping is also differentiable (on ). This theorem can be generalized to the infinite-dimensional case: Let and be complete normed spaces, let be an open set and let be a continuously-differentiable mapping. If is an invertible element in the space of bounded linear operators ( is the Fréchet derivative), , then there exists neighbourhoods and of and in and respectively, such that the mapping and its inverse mapping are continuously-differentiable homeomorphisms.

References

[1]	A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)

Comments

The assertions in the last paragraph of the main article are known under the (collective) name of inverse-function theorem.

Nowadays the term "function" is usually reserved for those relations that are single-valued, and the term "mapping" is one of its synonyms. When this is done, only bijections (one-to-one onto functions) have inverses that are functions. In all other cases, the inverse relation (called a many-valued function in the main article) is not a function unless, as is sometimes done, it is regarded as being set-valued. Then arises the important but simple distinction between a singleton set and its unique element.