# Step function

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.

## Definition and first consequences

A function
is called a **step function** if it can be written as

- real numbers

where and 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

## Examples

- A constant function is a trivial example of a step function. Then there is only one interval,
- The sign function which is −1 for negative numbers and +1 for positive numbers, and is the simplest non-constant step function.
- The Heaviside function
*H*(*x*), which is 0 for negative numbers and 1 for positive numbers, is equivalent to the sign function, up to a shift and scale of range ( ). 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 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[1] also define step functions 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 for in the above definition of the step function are disjoint and their union is the real line, then
- The definite integral of a step function is a piecewise linear function.
- 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]
- A discrete random variable is defined as a random variable whose cumulative distribution function is piecewise constant.[3]

## See also

## References

- 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) - Weir, Alan J. "3".
*Lebesgue integration and measure*. Cambridge University Press, 1973. ISBN 0-521-09751-7. - Bertsekas, Dimitri P. (2002).
*Introduction to Probability*. Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν. Belmont, Mass.: Athena Scientific. ISBN 188652940X. OCLC 51441829.

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