# 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.

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

## Definition and first consequences

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

${\displaystyle f(x)=\sum \limits _{i=0}^{n}\alpha _{i}\chi _{A_{i}}(x)\,}$  for all real numbers ${\displaystyle x}$

where ${\displaystyle n\geq 0}$  and ${\displaystyle \alpha _{i}}$  are real numbers, ${\displaystyle A_{i}}$  are intervals, and ${\displaystyle \chi _{A}}$  is the indicator function of ${\displaystyle A{\text{:}}}$

${\displaystyle \chi _{A}(x)={\begin{cases}1&{\text{if }}x\in A,\\0&{\text{if }}x\notin A.\\\end{cases}}}$

In this definition, the intervals ${\displaystyle A_{i}}$  can be assumed to have the following two properties:

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

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

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

can be written as

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

## Examples

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

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 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 ${\displaystyle A_{i},}$  for ${\displaystyle i=0,1,\dots ,n}$  in the above definition of the step function are disjoint and their union is the real line, then ${\displaystyle f(x)=\alpha _{i}}$  for all ${\displaystyle x\in A_{i}.}$
• The definite integral of a step function is a piecewise linear function.
• The Lebesgue integral of a step function ${\displaystyle \textstyle f=\sum \limits _{i=0}^{n}\alpha _{i}\chi _{A_{i}}}$  is ${\displaystyle \textstyle \int f\,dx=\sum \limits _{i=0}^{n}\alpha _{i}\ell (A_{i}),\,}$  where ${\displaystyle \textstyle \ell (A)}$  is the length of the interval ${\displaystyle A,}$  and it is assumed here that all intervals ${\displaystyle A_{i}}$  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]