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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     are real numbers,   are intervals, and   (sometimes written as  ) 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 an important step function, and 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     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.
  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.