Integer-valued function

In mathematics, an integer-valued function is a function whose values are integers. In other words, it is a function that assigns an integer to each member of its domain.

The

discontinuities or is constant. On the other hand, on discrete and other totally disconnected spaces integer-valued functions have roughly the same importance as real-valued functions

have on non-discrete spaces.

Any function with

non-negative

integer values is a partial case of an integer-valued function.

Examples

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

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

Algebraic properties

On an arbitrary

algebra over the ring

Z

of integers. Since the latter is an ordered ring, the functions form a partially ordered ring

:

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

Uses

Graph theory and algebra

Integer-valued functions are ubiquitous in

metric

.

Integer-valued polynomials are important in ring theory.

Mathematical logic and computability theory

In

μ-recursive functions represent integer-valued functions of several natural variables or, in other words, functions on

N n

. Gödel numbering, defined on well-formed formulae of some formal language

, is a natural-valued function.

Computability theory is essentially based on natural numbers and natural (or integer) functions on them.

Number theory

In number theory, many arithmetic functions are integer-valued.

Computer science

In

functions return values of integer type

due to simplicity of implementation.

References

https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/SurveyIntegerValuedEntireFunctions.pdf

Function
$x \mapsto f (x)$
History of the function concept
Examples of domains and codomains
$X$ → $\mathbb {B}$ , $\mathbb {B}$ → $X$ , $\mathbb {B} ^{n}$ → $X$ $X$ → $\mathbb {Z}$ , $\mathbb {Z}$ → $X$ $X$ → $\mathbb {R}$ , $\mathbb {R}$ → $X$ , $\mathbb {R} ^{n}$ → $X$ $X$ → $\mathbb {C}$ , $\mathbb {C}$ → $X$ , $\mathbb {C} ^{n}$ → $X$
Classes/properties
Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions
Restriction Composition λ Inverse
Generalizations
Binary relation Partial Multivalued Implicit Space
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