Step function explained

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

f\colon R → R is called a step function if it can be written as f(x) = \sum\limits ^{n} _{i =0} \alpha _{i} \chi _{A _{i}} (x) , for all real numbers x

where

n\ge 0 , \alpha _{i} are real numbers, A _{i} are intervals, and \chi _{A} is the indicator function of A :\chi _{A(x)} = \begin{cases} 1 & if x \in A \\ 0 & 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} \cap A _{j} = \emptyset for i ≠ j

1. The union of the intervals is the entire real line:

c up ^{n} _{i =0} A _{i} = 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 \chi _{[-5,} + 3 \chi _{(0,}

can be written as

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

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

• A constant function is a trivial example of a step function. Then there is only one interval,

A _{0=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 = ( 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.

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.

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 , ... , n in the above definition of the step function are disjoint and their union is the real line, then f(x) = \alpha _{i} for all x\in A _{i.}

• The definite integral of a step function is a piecewise linear function.
• The Lebesgue integral of a step function

style f = \sum ^{n} _{i =0} \alpha _{i} \chi _{A _{i}} is style \int fdx = \sum ^{n} _{i =0} \alpha _{i} \ell(A _{i),} where \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.^[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

Notes and References

1. Web site: Step Function.
2. Web site: Step Functions - Mathonline.
3. Web site: Mathwords: Step Function.
4. https://study.com/academy/lesson/step-function-definition-equation-examples.html
5. Web site: Step Function.
6. Book: Bachman, Narici, Beckenstein . Fourier and Wavelet Analysis . Springer, New York, 2000 . 0-387-98899-8 . Example 7.2.2. 5 April 2002 .
7. Book: Weir, Alan J . Lebesgue integration and measure . 10 May 1973. Cambridge University Press, 1973 . 0-521-09751-7 . 3.
8. Book: Bertsekas, Dimitri P.. Introduction to Probability. Dimitri Bertsekas. 2002. Athena Scientific. Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν.. 188652940X. Belmont, Mass.. 51441829.