Step function

A function $f\backslash colon\; \backslash mathbb\{R\}\; \backslash rightarrow\; \backslash mathbb\{R\}$ is called a step function if it can be written as {{Citation needed|date=September 2009}}

:$f(x)\; =\; \backslash sum\backslash limits\_\{i=0\}^n\; \backslash alpha\_i\; \backslash chi\_\{A\_i\}(x)$, for all real numbers $x$

where $n\backslash ge\; 0$, $\backslash alpha\_i$ are real numbers, $A\_i$ are intervals, and $\backslash chi\_A$ is the indicator function of $A$:

:$\backslash chi\_A(x)\; =\; \backslash begin\{cases\}$

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: $A\_i\; \backslash cap\; A\_j\; =\; \backslash emptyset$ for $i\; \backslash neq\; j$
2. The union of the intervals is the entire real line: $\backslash bigcup\_\{i=0\}^n\; A\_i\; =\; \backslash mathbb\; R.$

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

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

can be written as

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

Sometimes, the intervals are required to be right-open{{Cite web|url=http://mathworld.wolfram.com/StepFunction.html|title = Step Function}} or allowed to be singleton.{{Cite web|url=http://mathonline.wikidot.com/step-functions|title = Step Functions - Mathonline}} The condition that the collection of intervals must be finite is often dropped, especially in school mathematics,{{Cite web|url=https://www.mathwords.com/s/step_function.htm|title=Mathwords: Step Function}}{{Cite web | title=Archived copy | url=https://study.com/academy/lesson/step-function-definition-equation-examples.html | archive-url=https://web.archive.org/web/20150912010951/http://study.com:80/academy/lesson/step-function-definition-equation-examples.html | access-date=2024-12-16 | archive-date=2015-09-12}}{{Cite web|url=https://www.varsitytutors.com/hotmath/hotmath_help/topics/step-function|title = Step Function}} though it must still be locally finite, resulting in the definition of piecewise constant functions.

Image:Dirac distribution CDF.svg is an often-used step function.]]

• A constant function is a trivial example of a step function. Then there is only one interval, $A\_0=\backslash mathbb\; R.$
• The sign function {{math|sgn(x)}}, which is −1 for negative numbers and +1 for positive numbers, and is the simplest non-constant step function.
• The Heaviside function {{math|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 ($H\; =\; (\backslash sgn\; +\; 1)/2$). It is the mathematical concept behind some test signals, such as those used to determine the step response of a dynamical system.

File:Rectangular function.svg, the next simplest step function.]]

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

• 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{{Cite book | author=Bachman, Narici, Beckenstein | title=Fourier and Wavelet Analysis | publisher=Springer, New York, 2000 | isbn=0-387-98899-8 | chapter =Example 7.2.2| date=5 April 2002 }} also define step functions with an infinite number of intervals.

• 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 $i=0,\; 1,\; \backslash dots,\; n$ in the above definition of the step function are disjoint and their union is the real line, then $f(x)=\backslash alpha\_i$ for all $x\backslash in\; A\_i.$
• The definite integral of a step function is a piecewise linear function.
• The Lebesgue integral of a step function $\backslash textstyle\; f\; =\; \backslash sum\_\{i=0\}^n\; \backslash alpha\_i\; \backslash chi\_\{A\_i\}$ is $\backslash textstyle\; \backslash int\; f\backslash ,dx\; =\; \backslash sum\_\{i=0\}^n\; \backslash alpha\_i\; \backslash ell(A\_i),$ where $\backslash ell(A)$ 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.{{Cite book | author=Weir, Alan J | title=Lebesgue integration and measure | date= 10 May 1973| publisher=Cambridge University Press, 1973 | isbn=0-521-09751-7 |chapter= 3}}
• A discrete random variable is sometimes defined as a random variable whose cumulative distribution function is piecewise constant.{{Cite book|title=Introduction to Probability|last=Bertsekas|author-link=Dimitri Bertsekas|first=Dimitri P.|date=2002|publisher=Athena Scientific|others=Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν.|isbn=188652940X|location=Belmont, Mass.|oclc=51441829}} 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.

• Crenel function
• Piecewise
• Sigmoid function
• Simple function
• Step detection
• Heaviside step function
• Piecewise-constant valuation

{{Reflist}}

{{DEFAULTSORT:Step Function}}

Category:Special functions