In the mathematical field of real analysis, a simple function is a (sufficiently 'nice' - see below for the formal definition) real-valued function over a subset of the real line which attains only a finite number of values. Some authors also require simple functions to be measurable; as used in practice, they invariably are.
A basic example of a simple function is the floor function over the half-open interval [1,9), whose only values are {1,2,3,4,5,6,7,8}. A more advanced example is the Dirichlet function over the real line, which takes the value 1 if x is rational and 0 otherwise. (Thus the "simple" of "simple function" has a technical meaning somewhat at odds with common language.) Note also that all step functions are simple.
Simple functions are used as a first stage in the development of theories of integration, such as the Lebesgue integral, because it is very easy to create a definition of an integral for a simple function, and also, it is straightforward to approximate more general functions by sequences of simple functions.
Read more about Simple Function: Definition, Properties of Simple Functions, Integration of Simple Functions, Relation To Lebesgue Integration
Famous quotes containing the words simple and/or function:
“The prostitute is the scapegoat for everyone’s sins, and few people care whether she is justly treated or not. Good people have spent thousands of pounds in efforts to reform her, poets have written about her, essayists and orators have made her the subject of some of their most striking rhetoric; perhaps no class of people has been so much abused, and alternatively sentimentalized over as prostitutes have been but one thing they have never yet had, and that is simple legal justice.”
—Alison Neilans. “Justice for the Prostitute—Lady Astor’s Bill,” Equal Rights (September 19, 1925)
“Science has fulfilled her function when she has ascertained and enunciated truth.”
—Thomas Henry Huxley (1825–95)