Step function

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In mathematics, a function on the real numbers is called a step function (or staircase 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.

Example of a step function (the red graph). This particular step function is right-continuous.

Definition and first consequences[edit]

A function is called a step function if it can be written as[citation needed]

, for all real numbers

where , are real numbers, are intervals, and is the indicator function of :

In this definition, the intervals can be assumed to have the following two properties:

  1. The intervals are pairwise disjoint: for
  2. The union of the intervals is the entire real line:

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

can be written as

Variations in the definition[edit]

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[edit]

The Heaviside step function is an often-used step function.
  • A constant function is a trivial example of a step function. Then there is only one interval,
  • 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 (). 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 next simplest step function.

Non-examples[edit]

  • 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[edit]

  • 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 for in the above definition of the step function are disjoint and their union is the real line, then for all
  • The definite integral of a step function is a piecewise linear function.
  • The Lebesgue integral of a step function is where is the length of the interval , and it is assumed here that all intervals 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[edit]

References[edit]

  1. ^ http://mathworld.wolfram.com/StepFunction.html
  2. ^ http://mathonline.wikidot.com/step-functions
  3. ^ https://www.mathwords.com/s/step_function.htm
  4. ^ https://study.com/academy/lesson/step-function-definition-equation-examples.html
  5. ^ https://www.varsitytutors.com/hotmath/hotmath_help/topics/step-function
  6. ^ a b Bachman, Narici, Beckenstein (5 April 2002). "Example 7.2.2". Fourier and Wavelet Analysis. Springer, New York, 2000. ISBN 0-387-98899-8.CS1 maint: multiple names: authors list (link)
  7. ^ Weir, Alan J (10 May 1973). "3". Lebesgue integration and measure. Cambridge University Press, 1973. ISBN 0-521-09751-7.
  8. ^ Bertsekas, Dimitri P. (2002). Introduction to Probability. Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν. Belmont, Mass.: Athena Scientific. ISBN 188652940X. OCLC 51441829.