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.
Inverse function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inverse_function&oldid=29492