# Step function

From Infogalactic: the planetary knowledge core

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.

## Contents

## Definition and first consequences

A function is called a **step function** if it can be written as^{[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:

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

- A constant function is a trivial example of a step function. Then there is only one interval,
- The Heaviside function
*H*(*x*) is an important step function. It is the mathematical concept behind some test signals, such as those used to determine the step response of a dynamical system.

- The rectangular function, the normalized boxcar function, is the next simplest step function, and is used to model a unit pulse.

### 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

- The sum and product of two step functions is again a step function. The product of a step function with a number is also a step function. As such, the step functions form an algebra over the real numbers.
- A step function takes only a finite number of values. If the intervals in the above definition of the step function are disjoint and their union is the real line, then for all
- The Lebesgue integral of a step function is where is the length of the interval and it is assumed here that all intervals have finite length. In fact, this equality (viewed as a definition) can be the first step in constructing the Lebesgue integral.
^{[2]}

## See also

- Unit Step function
- Crenel function
- Simple function
- Piecewise defined function
- Sigmoid function
- Step detection

## References

- ↑ for example see: Bachman, Narici, Beckenstein. "Example 7.2.2".
*Fourier and Wavelet Analysis*. Springer, New York, 2000. ISBN 0-387-98899-8.CS1 maint: multiple names: authors list (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Weir, Alan J. "3".
*Lebesgue integration and measure*. Cambridge University Press, 1973. ISBN 0-521-09751-7.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>