Movatterモバイル変換

Integer-valued function

From Wikipedia, the free encyclopedia

This articlerelies largely or entirely on asingle source. Relevant discussion may be found on thetalk page. Please helpimprove this article byintroducing citations to additional sources.
Find sources: "Integer-valued function" – news ·newspapers ·books ·scholar ·JSTOR(June 2013)

The floor function on real numbers. Its discontinuities are pictured with white discs outlines with blue circles.

Function
x ↦f (x)
History of the function concept
Types bydomain andcodomain
`X` →𝔹 𝔹 →`X` 𝔹ⁿ →`X` `X` →ℤ ℤ →`X` `X` →ℝ ℝ →`X` ℝⁿ →`X` `X` →ℂ ℂ →`X` ℂⁿ →`X`
Classes/properties
Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions
Restriction Composition λ Inverse
Generalizations
Relation (Binary relation) Set-valued Multivalued Partial Implicit Space Higher-order Morphism Functor
List of specific functions
v t e

Inmathematics, aninteger-valued function is afunction whose values areintegers. In other words, it is a function that assigns an integer to each member of itsdomain.

Thefloor and ceiling functions are examples of integer-valuedfunctions of a real variable, but onreal numbers and, generally, on (non-disconnected)topological spaces integer-valued functions are not especially useful. Any such function on aconnected space either hasdiscontinuities or isconstant. On the other hand, ondiscrete and othertotally disconnected spaces integer-valued functions have roughly the same importance asreal-valued functions have on non-discrete spaces.

Any function withnatural, ornon-negative integer values is a partial case of an integer-valued function.

Examples

[edit]

Integer-valued functions defined on the domain of all real numbers include the floor and ceiling functions, theDirichlet function, thesign function and theHeaviside step function (except possibly at 0).

Integer-valued functions defined on the domain of non-negative real numbers include theinteger square root function and theprime-counting function.

Algebraic properties

[edit]

On an arbitrarysetX, integer-valued functions form aring withpointwise operations of addition and multiplication,^[1] and also analgebra over the ringZ of integers. Since the latter is anordered ring, the functions form apartially ordered ring:

f\leq g\quad \iff \quad \forall x:f(x)\leq g(x).

Uses

[edit]

Graph theory and algebra

[edit]

Integer-valued functions are ubiquitous ingraph theory. They also have similar uses ingeometric group theory, wherelength function represents the concept ofnorm, andword metric represents the concept ofmetric.

Integer-valued polynomials are important inring theory.