Step function

Informally speaking, a step function is a piecewise constant function having only finitely many pieces.

is called a step function if it can be written as [citation needed] where

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.

An example of step functions (the red graph). In this function, each constant subfunction with a function value α i ( i = 0, 1, 2, ...) is defined by an interval A i and intervals are distinguished by points x j ( j = 1, 2, ...). This particular step function is right-continuous .
The Heaviside step function is an often-used step function.
The rectangular function , the next simplest step function.