# 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 $f\colon \mathbb {R} \rightarrow \mathbb {R}$ is called a step function if it can be written as[citation needed]

$f(x)=\sum \limits _{i=0}^{n}\alpha _{i}\chi _{A_{i}}(x)\ \forall$ real numbers $x$ where $n\geq 0$ and $\alpha _{i}$ are real numbers, $A_{i}$ are intervals, and $\chi _{A}$ is the indicator function of $A{\text{:}}$ $\chi _{A}(x)={\begin{cases}1&{\text{if }}x\in A,\\0&{\text{if }}x\notin A.\\\end{cases}}$ In this definition, the intervals $A_{i}$ can be assumed to have the following two properties:

1. The intervals are pairwise disjoint: $A_{i}\cap A_{j}=\emptyset$ for $i\neq j$ 2. The union of the intervals is the entire real line: $\bigcup _{i=0}^{n}A_{i}=\mathbb {R} .$ 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

• A constant function is a trivial example of a step function. Then there is only one interval, $A_{0}=\mathbb {R} .$ • The sign function $\operatorname {sgn} (x),$ 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 ($H=(\operatorname {sgn} +1)/2$ ). It is the mathematical concept behind some test signals, such as those used to determine the step response of a dynamical system.

### 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 also define step functions with an infinite number of intervals.

## 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 $A_{i},$ for $i=0,1,\dots ,n$ in the above definition of the step function are disjoint and their union is the real line, then $f(x)=\alpha _{i}\ \forall \ x\in A_{i}.$ • The definite integral of a step function is a piecewise linear function.
• The Lebesgue integral of a step function $\textstyle f=\sum \limits _{i=0}^{n}\alpha _{i}\chi _{A_{i}}$ is $\textstyle \int f\,dx=\sum \limits _{i=0}^{n}\alpha _{i}\ell (A_{i}),\,$ where $\textstyle \ell (A)$ 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.
• A discrete random variable is defined as a random variable whose cumulative distribution function is piecewise constant.