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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.

Contents

Definition and first consequencesEdit

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

  for all real numbers  

where   and   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

 

ExamplesEdit

 
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   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-examplesEdit

  • 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[1] also define step functions with an infinite number of intervals.[1]

PropertiesEdit

  • 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.[2]
  • A discrete random variable is defined as a random variable whose cumulative distribution function is piecewise constant.[3]

See alsoEdit

ReferencesEdit

  1. ^ a b Bachman, Narici, Beckenstein. "Example 7.2.2". Fourier and Wavelet Analysis. Springer, New York, 2000. ISBN 0-387-98899-8.CS1 maint: Multiple names: authors list (link)
  2. ^ Weir, Alan J. "3". Lebesgue integration and measure. Cambridge University Press, 1973. ISBN 0-521-09751-7.
  3. ^ Bertsekas, Dimitri P. (2002). Introduction to Probability. Tsitsiklis, John N., Τσιτσικλής, Γιάννης Ν. Belmont, Mass.: Athena Scientific. ISBN 188652940X. OCLC 51441829.