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 Americans never use the word peasant, because they have no idea of the class which that term denotes; the ignorance of more remote ages, the simplicity of rural life, and the rusticity of the villager have not been preserved among them; and they are alike unacquainted with the virtues, the vices, the coarse habits, and the simple graces of an early stage of civilization.”
—Alexis de Tocqueville (1805–1859)
“For me being a poet is a job rather than an activity. I feel I have a function in society, neither more nor less meaningful than any other simple job. I feel it is part of my work to make poetry more accessible to people who have had their rights withdrawn from them.”
—Jeni Couzyn (b. 1942)