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:
“Geez, if I could get through to you, kiddo, that depression is not sobbing and crying and giving vent, it is plain and simple reduction of feeling. Reduction, see? Of all feeling. People who keep stiff upper lips find that it’s damn hard to smile.”
—Judith Guest (b. 1936)
“As a medium of exchange,... worrying regulates intimacy, and it is often an appropriate response to ordinary demands that begin to feel excessive. But from a modernized Freudian view, worrying—as a reflex response to demand—never puts the self or the objects of its interest into question, and that is precisely its function in psychic life. It domesticates self-doubt.”
—Adam Phillips, British child psychoanalyst. “Worrying and Its Discontents,” in On Kissing, Tickling, and Being Bored, p. 58, Harvard University Press (1993)