Integervalued function
In mathematics, an integervalued 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.
Function  

x ↦ f (x)  
Examples by domain and codomain  


Classes/properties  
Constant · Identity · Linear · Polynomial · Rational · Algebraic · Analytic · Smooth · Continuous · Measurable · Injective · Surjective · Bijective  
Constructions  
Restriction · Composition · λ · Inverse  
Generalizations  
Partial · Multivalued · Implicit  
Floor and ceiling functions are examples of an integervalued function of a real variable, but on real numbers and generally, on (nondisconnected) topological spaces integervalued functions are not especially useful. Any such function on a connected space either has discontinuities or is constant. On the other hand, on discrete and other totally disconnected spaces integervalued functions have roughly the same importance as realvalued functions have on nondiscrete spaces.
Any function with natural, or nonnegative integer values is a partial case of integervalued function.
Examples
Integervalued 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).
Integervalued functions defined on the domain of nonnegative real numbers include the integer square root function and the primecounting function.
Algebraic properties
On an arbitrary set X, integervalued functions form a ring with pointwise operations of addition and multiplication, and also an algebra over the ring Z of integers. Since the latter is an ordered ring, the functions form a partially ordered ring:
Uses
Graph theory and algebra
Integervalued functions are ubiquitous in graph theory. They also have similar uses in geometric group theory, where length function represents the concept of norm, and word metric represents the concept of metric.
Integervalued polynomials are important in ring theory.
Mathematical logic and computability theory
In mathematical logic such concepts as a primitive recursive function and a μrecursive function represent integervalued functions of several natural variables or, in other words, functions on N^{n}. Gödel numbering, defined on wellformed formulae of some formal language, is a naturalvalued 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 integervalued.
Computer science
In computer programming many functions return values of integer type due to simplicity of implementation.