Linear combination of indicator functions of real intervals
In mathematics, a function on the real numbers is called a step 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
- , for all real numbers
where , are real numbers, are intervals, and is the indicator function of :
In this definition, the intervals can be assumed to have the following two properties:
- The intervals are pairwise disjoint: for
- 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
Variations in the definition
Sometimes, the intervals are required to be right-open or allowed to be singleton. The condition that the collection of intervals must be finite is often dropped, especially in school mathematics, though it must still be locally finite, resulting in the definition of piecewise constant functions.