Step function

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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.

File:StepFunctionExample.png

An example of step functions (the red graph). In this function, each constant subfunction with a function value α_i (i = 0, 1, 2, ...) is defined by an interval A_i and intervals are distinguished by points x_j (j = 1, 2, ...). This particular step function is right-continuous.

Definition and first consequencesEdit

A function <math>f\colon \mathbb{R} \rightarrow \mathbb{R}</math> is called a step function if it can be written as Template:Citation needed

<math>f(x) = \sum\limits_{i=0}^n \alpha_i \chi_{A_i}(x)</math>, for all real numbers <math>x</math>

where <math>n\ge 0</math>, <math>\alpha_i</math> are real numbers, <math>A_i</math> are intervals, and <math>\chi_A</math> is the indicator function of <math>A</math>:

<math>\chi_A(x) = \begin{cases}

 1 & \text{if } x \in A \\
 0 & \text{if } x \notin A \\
\end{cases}</math>

In this definition, the intervals <math>A_i</math> can be assumed to have the following two properties:

The intervals are pairwise disjoint: <math>A_i \cap A_j = \emptyset</math> for <math>i \neq j</math>
The union of the intervals is the entire real line: <math>\bigcup_{i=0}^n A_i = \mathbb R.</math>

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

<math>f = 0\chi_{(-\infty, -5)} +4 \chi_{[-5, 0]} +7 \chi_{(0, 1)} + 3 \chi_{[1, 6)}+0\chi_{[6, \infty)}.</math>

Variations in the definitionEdit

Sometimes, the intervals are required to be right-open<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> or allowed to be singleton.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> The condition that the collection of intervals must be finite is often dropped, especially in school mathematics,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> though it must still be locally finite, resulting in the definition of piecewise constant functions.

ExamplesEdit

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, <math>A_0=\mathbb R.</math>
The sign function Template:Math, which is −1 for negative numbers and +1 for positive numbers, and is the simplest non-constant step function.
The Heaviside function Template:Math, 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 (<math>H = (\sgn + 1)/2</math>). 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.

The rectangular function, the normalized boxcar function, is used to model a unit pulse.

Non-examplesEdit

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<ref name=bachman_narici_beckenstein>Template:Cite book</ref> also define step functions with an infinite number of intervals.<ref name=bachman_narici_beckenstein />

PropertiesEdit

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 <math>A_i,</math> for <math>i=0, 1, \dots, n</math> in the above definition of the step function are disjoint and their union is the real line, then <math>f(x)=\alpha_i</math> for all <math>x\in A_i.</math>
The definite integral of a step function is a piecewise linear function.
The Lebesgue integral of a step function <math>\textstyle f = \sum_{i=0}^n \alpha_i \chi_{A_i}</math> is <math>\textstyle \int f\,dx = \sum_{i=0}^n \alpha_i \ell(A_i),</math> where <math>\ell(A)</math> is the length of the interval <math>A</math>, and it is assumed here that all intervals <math>A_i</math> have finite length. In fact, this equality (viewed as a definition) can be the first step in constructing the Lebesgue integral.<ref>Template:Cite book</ref>
A discrete random variable is sometimes defined as a random variable whose cumulative distribution function is piecewise constant.<ref name=":0">Template:Cite book</ref> 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.

ReferencesEdit

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