In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combinationofindicator functionsofintervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces.
A function is called a step function if it can be written as [citation needed]
where ,
are real numbers,
are intervals, and
is the indicator functionof
:
In this definition, the intervals can be assumed to have the following two properties:
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
Sometimes, the intervals are required to be right-open[1] or allowed to be singleton.[2] The condition that the collection of intervals must be finite is often dropped, especially in school mathematics,[3][4][5] though it must still be locally finite, resulting in the definition of piecewise constant functions.
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