Step function

 ( Special Functions ) 

In mathematics, a function on the is called a step function if it can be written as a finite set linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces.

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Definition and first consequences A function $f\backslash colon\; \backslash mathbb\{R\}\; \backslash rightarrow\; \backslash mathbb\{R\}$ is called a step function if it can be written as

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

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

$\backslash chi\_\{A\_\{i\}\}(x)\; =\; \backslash begin\{cases\}$

 1 & \text{if } x \in A_{i} \\
 0 & \text{if } x \notin A_{i} \\
\end{cases}
     

In this definition, the intervals $A\_i$ can be assumed to have the following two properties:

1. The intervals are disjoint sets: $A\_i\; \backslash cap\; A\_j\; =\; \backslash emptyset$ 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

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

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

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Examples

• 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 , which is −1 for negative numbers and +1 for positive numbers, and is the simplest non-constant step function.
• The Heaviside function , 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.
• The rectangular function, the normalized boxcar function, is used to model a unit pulse.

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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
  Bachman, Narici, Beckenstein (2002). 9780387988993, Springer, New York, 2000. ISBN 9780387988993
  also define step functions with an infinite number of intervals.

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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 $i=0,\; 1,\; \backslash dots,\; n$ 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.
  Weir, Alan J (1973). 9780521097512, Cambridge University Press, 1973. ISBN 9780521097512
• A discrete random variable is sometimes defined as a random variable whose cumulative distribution function is piecewise constant.
  Dimitri P. Bertsekas (2025). 188652940X, Athena Scientific. 188652940X
  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.

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See also

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

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