In mathematics, afunction on thereal numbers is called astep function if it can be written as afinitelinear combination ofindicator functions ofintervals. Informally speaking, a step function is apiecewiseconstant function having only finitely many pieces.

A function${\displaystyle f:\mathbb{R}\to \mathbb{R}}$ is called astep function if it can be written as^{[citation needed]}

${\displaystyle f(x)=\sum _{i=0}^n{\alpha }_i{\chi }_{A_i}(x)}$, for all real numbers${\displaystyle x}$

where${\displaystyle n\ge 0}$,${\displaystyle {\alpha }_i}$ are real numbers,${\displaystyle A_i}$ are intervals, and${\displaystyle {\chi }_A}$ is theindicator function of${\displaystyle A}$:

In this definition, the intervals${\displaystyle A_i}$ can be assumed to have the following two properties:

1. The intervals arepairwise disjoint:${\displaystyle A_i\cap A_j=\mathrm{\varnothing }}$ for${\displaystyle i\ne j}$
2. Theunion of the intervals is the entire real line:${\displaystyle \bigcup _{i=0}^n{A}_i=\mathbb{R}.}$

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

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

can be written as

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

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 belocally finite, resulting in the definition of piecewise constant functions.

• Aconstant function is a trivial example of a step function. Then there is only one interval,${\displaystyle A_0=\mathbb{R}.}$
• Thesign functionsgn(x), which is −1 for negative numbers and +1 for positive numbers, and is the simplest non-constant step function.
• TheHeaviside functionH(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 (${\displaystyle H=(\mathrm{sgn}+1)/2}$). It is the mathematical concept behind some testsignals, such as those used to determine thestep response of adynamical system.

• Therectangular function, the normalizedboxcar function, is used to model a unit pulse.

• Theinteger 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]

• 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 analgebra over the real numbers.
• A step function takes only a finite number of values. If the intervals${\displaystyle A_i,}$ for${\displaystyle i=0,1,\dots ,n}$ in the above definition of the step function are disjoint and their union is the real line, then${\displaystyle f(x)={\alpha }_i}$ for all${\displaystyle x\in A_i.}$
• Thedefinite integral of a step function is apiecewise linear function.
• TheLebesgue integral of a step function${\displaystyle f=\sum _{i=0}^n{\alpha }_i{\chi }_{A_i}}$ is${\displaystyle \int fdx=\sum _{i=0}^n{\alpha }_i\ell (A_i),}$ where${\displaystyle \ell (A)}$ is the length of the interval${\displaystyle A}$, and it is assumed here that all intervals${\displaystyle A_i}$ have finite length. In fact, this equality (viewed as a definition) can be the first step in constructing the Lebesgue integral.^[7]
• Adiscrete random variable is sometimes defined as arandom variable whosecumulative 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.

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

• Special functions

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