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[[Image:Floor function.svg|thumb|right|288px|The floor function on real numbers. Its discontinuities are pictured with white discs outlines with blue circles.]]
{{Functions}}
In mathematics, an '''integer-valued function''' is a [[function (mathematics)|function]] whose [[codomain|values]] are [[integer]]s.  In other words, it is a function that assigns an integer to each member of its [[domain of a function|domain]].

[[Floor and ceiling functions]] are examples of an integer-valued [[function of a real variable]], but on [[real number]]s and generally, on (non-disconnected) [[topological space]]s integer-valued functions are not especially useful. Any such function on a [[connected space]] either has [[discontinuity (mathematics)|discontinuities]] or is [[constant function|constant]]. On the other hand, on [[discrete space|discrete]] and other [[totally disconnected space]]s integer-valued functions have roughly the same importance as [[real-valued function]]s have on non-discrete spaces.

Any function with [[natural number|natural]], or [[non-negative]] integer values is a partial case of 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 [[set (mathematics)|set]] {{mvar|X}}, integer-valued functions form a [[ring (mathematics)|ring]] with [[pointwise]] operations of [[addition]] and  [[multiplication]], and also an [[algebra (ring theory)|algebra]] over the ring {{math|'''Z'''}} of integers. Since the latter is an [[ordered ring]], the functions form a [[partially ordered ring]]:
:<math>f \le g \quad\iff\quad \forall x: f(x) \le g(x).</math>

== Uses ==
=== Graph theory and algebra ===
Integer-valued functions are ubiquitous in [[graph theory]]. They also have similar uses in [[geometric group theory]], where ''[[length function]]'' represents the concept of [[norm (mathematics)|norm]], and ''[[word metric]]'' represents the concept of [[metric (mathematics)|metric]].

[[Integer-valued polynomial]]s are important in [[ring theory]].

=== Mathematical logic and computability theory ===
In [[mathematical logic]] such concepts as a [[primitive recursive function]] and a [[μ-recursive function]] represent integer-valued functions of several natural variables or, in other words, functions on {{math|'''N'''[[coordinate space|<sup>''n''</sup>]]}}. [[Gödel numbering]], defined on [[well-formed formula]]e 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 function]]s are integer-valued.

=== Computer science ===
In [[computer programming]] many [[function (programming)|functions]] return values of [[Integer (computer science)|integer type]] due to simplicity of implementation.

== See also ==
<!-- * [[Function of an integer variable]] (also known as a doubly-[[infinity|infinite]] sequence), the dual concept // obsoleted by the sidebar? -->
* [[Semi-continuity]]
* [[Rank (disambiguation) #Mathematics]]
* [[Grade (disambiguation) #In mathematics]]


[[Category:Types of functions]]
[[Category:Ring theory]]
[[Category:Geometric group theory]]
[[Category:Graph theory]]
[[Category:Mathematical logic]]
[[Category:Discrete mathematics]]