Step function

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In mathematics, a function on the real numbers is called a step function (or staircase function) if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces.

Example of a step function (the red graph). This particular step function is right-continuous.

Definition and first consequences

A function is called a step function if it can be written as {{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

for all real numbers

where are real numbers, are intervals, and (sometimes written as ) is the indicator function of :

In this definition, the intervals can be assumed to have the following two properties:

  1. The intervals are disjoint, for
  2. The union of the intervals is the entire real line,

Indeed, if that is not the case to start with, a different set of intervals can be picked for which these assumptions hold. For example, the step function

can be written as

Examples

The Heaviside step function is an often used step function.
The rectangular function, the next simplest step function.

Non-examples

  • The integer part function is not a step function according to the definition of this article, since it has an infinite number of intervals. However, some authors define step functions also with an infinite number of intervals.[1]

Properties

See also

References

  1. for example see: {{#invoke:citation/CS1|citation |CitationClass=book }}
  2. {{#invoke:citation/CS1|citation |CitationClass=book }}