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 [math]\displaystyle{ f\colon \mathbb{R} \rightarrow \mathbb{R} }[/math] is called a step function if it can be written as[citation needed]
- [math]\displaystyle{ f(x) = \sum\limits_{i=0}^n \alpha_i \chi_{A_i}(x) }[/math], for all real numbers [math]\displaystyle{ x }[/math]
where [math]\displaystyle{ n\ge 0 }[/math], [math]\displaystyle{ \alpha_i }[/math] are real numbers, [math]\displaystyle{ A_i }[/math] are intervals, and [math]\displaystyle{ \chi_A }[/math] is the indicator function of [math]\displaystyle{ A }[/math]:
- [math]\displaystyle{ \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]\displaystyle{ A_i }[/math] can be assumed to have the following two properties:
- The intervals are pairwise disjoint: [math]\displaystyle{ A_i \cap A_j = \emptyset }[/math] for [math]\displaystyle{ i \neq j }[/math]
- The union of the intervals is the entire real line: [math]\displaystyle{ \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
- [math]\displaystyle{ f = 4 \chi_{[-5, 1)} + 3 \chi_{(0, 6)} }[/math]
can be written as
- [math]\displaystyle{ 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 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
![](/wiki/images/thumb/d/d9/Dirac_distribution_CDF.svg/325px-Dirac_distribution_CDF.svg.png)
- A constant function is a trivial example of a step function. Then there is only one interval, [math]\displaystyle{ A_0=\mathbb R. }[/math]
- 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 ([math]\displaystyle{ 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.
![](/wiki/images/thumb/1/11/Rectangular_function.svg/250px-Rectangular_function.svg.png)
- 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 [math]\displaystyle{ A_i, }[/math] for [math]\displaystyle{ 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]\displaystyle{ f(x)=\alpha_i }[/math] for all [math]\displaystyle{ 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]\displaystyle{ \textstyle f = \sum_{i=0}^n \alpha_i \chi_{A_i} }[/math] is [math]\displaystyle{ \textstyle \int f\,dx = \sum_{i=0}^n \alpha_i \ell(A_i), }[/math] where [math]\displaystyle{ \ell(A) }[/math] is the length of the interval [math]\displaystyle{ A }[/math], and it is assumed here that all intervals [math]\displaystyle{ A_i }[/math] 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.
See also
- Crenel function
- Piecewise
- Sigmoid function
- Simple function
- Step detection
- Heaviside step function
- Piecewise-constant valuation
References
- ↑ "Step Function". http://mathworld.wolfram.com/StepFunction.html.
- ↑ "Step Functions - Mathonline". http://mathonline.wikidot.com/step-functions.
- ↑ "Mathwords: Step Function". https://www.mathwords.com/s/step_function.htm.
- ↑ https://study.com/academy/lesson/step-function-definition-equation-examples.html[bare URL]
- ↑ "Step Function". https://www.varsitytutors.com/hotmath/hotmath_help/topics/step-function.
- ↑ 6.0 6.1 Bachman, Narici, Beckenstein (5 April 2002). "Example 7.2.2". Fourier and Wavelet Analysis. Springer, New York, 2000. ISBN 0-387-98899-8.
- ↑ Weir, Alan J (10 May 1973). "3". Lebesgue integration and measure. Cambridge University Press, 1973. ISBN 0-521-09751-7.
- ↑ Bertsekas, Dimitri P. (2002). Introduction to Probability. Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν.. Belmont, Mass.: Athena Scientific. ISBN 188652940X. OCLC 51441829.
![]() | Original source: https://en.wikipedia.org/wiki/Step function.
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