# 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.

File:StepFunctionExample.png
Example of a step function (the red graph). This particular step function is right-continuous.

## Definition and first consequences

A function $f:\mathbb {R} \rightarrow \mathbb {R}$ is called a step function if it can be written as

$f(x)=\sum \limits _{i=0}^{n}\alpha _{i}\chi _{A_{i}}(x)\,$ for all real numbers $x$ where $n\geq 0,$ $\alpha _{i}$ are real numbers, $A_{i}$ are intervals, and $\chi _{A}\,$ is the indicator function of $A$ :

$\chi _{A}(x)=\left\{{\begin{matrix}1,&\mathrm {if} \;x\in A\\0,&\mathrm {otherwise} .\end{matrix}}\right.$ In this definition, the intervals $A_{i}$ can be assumed to have the following two properties:

• The intervals are disjoint, $A_{i}\cap A_{j}=\emptyset$ for $i\neq j$ • The union of the intervals is the entire real line, $\cup _{i=1}^{n}A_{i}=\mathbb {R} .$ 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

$f=4\chi _{[-5,1)}+3\chi _{(0,6)}\,$ can be written as

$f=0\chi _{(-\infty ,-5)}+4\chi _{[-5,0]}+7\chi _{(0,1)}+3\chi _{[1,6)}+0\chi _{[6,\infty )}.\,$ ## Examples

File:Dirac distribution CDF.svg
The Heaviside step function is an often used step function.
• A constant function is a trivial example of a step function. Then there is only one interval, $A_{0}=\mathbb {R} .$ • 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.
File:Rectangular function.svg
The rectangular function, the next simplest step function.

### 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 "steps".

## 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 $A_{i},$ $i=0,1,\dots ,n,$ in the above definition of the step function are disjoint and their union is the real line, then $f(x)=\alpha _{i}\,$ for all $x\in A_{i}.$ • The Lebesgue integral of a step function $f=\sum \limits _{i=0}^{n}\alpha _{i}\chi _{A_{i}}\,$ is $\int \!f\,dx=\sum \limits _{i=0}^{n}\alpha _{i}\ell (A_{i}),\,$ where $\ell (A)$ is the length of the interval $A,$ and it is assumed here that all intervals $A_{i}$ have finite length. In fact, this equality (viewed as a definition) can be the first step in constructing the Lebesgue integral.
• The derivative of a step function is the Dirac delta function
$\delta (x)={\begin{cases}+\infty ,&x=0\\0,&x\neq 0\end{cases}}$  