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Floor and ceiling functions are examples of an integer-valued function of a real variable, but on real numbers and generally, on (non-disconnected) topological spaces integer-valued 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 integer-valued functions have roughly the same importance as real-valued functions have on non-discrete spaces.
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).
On an arbitrary set X, integer-valued 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:
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, and word metric represents the concept of metric.
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 Nn. 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.