Step function

In mathematics, a function on the real numbers is called a step 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)

, 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)={\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:

The intervals are pairwise disjoint: $A_{i}\cap A_{j}=\emptyset$ for $i\neq j$
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 )}.

Variations in the definition

Sometimes, the intervals are required to be right-open^[1] or allowed to be singleton.^[2] The condition that the collection of intervals must be finite is often dropped, especially in school mathematics,^[3]^[4]^[5] though it must still be locally finite, resulting in the definition of piecewise constant functions.

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, $A_{0}=\mathbb {R} .$
The sign function $sgn(x)$ , 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.

The rectangular function, the next simplest step function.

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^[6] also define step functions with an infinite number of intervals.^[6]

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 $f(x)=\alpha _{i}$ for all $x\in A_{i}.$
The definite integral of a step function is a piecewise linear function.
The Lebesgue integral of a step function $\textstyle f=\sum _{i=0}^{n}\alpha _{i}\chi _{A_{i}}$ is $\textstyle \int f\,dx=\sum _{i=0}^{n}\alpha _{i}\ell (A_{i}),$ 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.^[7]
A discrete random variable is sometimes defined as a random variable whose cumulative distribution function is piecewise constant.^[8] In this case, it is locally a step function (globally, it may have an infinite number of steps). Usually however, any random variable with only countably many possible values is called a discrete random variable, in this case their cumulative distribution function is not necessarily locally a step function, as infinitely many intervals can accumulate in a finite region.