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.

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

Definition and first consequences

A function <math>f: \mathbb{R} \rightarrow \mathbb{R}</math> is called a step function if it can be written as

<math>f(x) = \sum\limits_{i=0}^n \alpha_i \chi_{A_i}(x)\,</math> for all real numbers <math>x</math>

where <math>n\ge 0,</math> <math>\alpha_i</math> are real numbers, <math>A_i</math> are intervals, and <math>\chi_A\,</math> is the indicator function of <math>A</math>:

<math>\chi_A(x) =

\left\{

 \begin{matrix}
   1, & \mathrm{if} \; x \in A \\ 
   0, & \mathrm{otherwise}. 
 \end{matrix}

\right. </math>

In this definition, the intervals <math>A_i</math> can be assumed to have the following two properties:

  • The intervals are disjoint, <math>A_i\cap A_j=\emptyset</math> for <math>i\ne j</math>
  • The union of the intervals is the entire real line, <math>\cup_{i=1}^n A_i=\mathbb R.</math>

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

<math>f = 4 \chi_{[-5, 1)} + 3 \chi_{(0, 6)}\,</math>

can be written as

<math>f = 0\chi_{(-\infty, -5)} +4 \chi_{[-5, 0]} +7 \chi_{(0, 1)} + 3 \chi_{[1, 6)}+0\chi_{[6, \infty)}.\,</math>

Examples

File:Dirac distribution CDF.svg
The Heaviside step function is an often used step function.
File:Rectangular function.svg
The rectangular function, the next simplest step function.

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 "steps".

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 <math>A_i,</math> <math>i=0, 1, \dots, n,</math> in the above definition of the step function are disjoint and their union is the real line, then <math>f(x)=\alpha_i\,</math> for all <math>x\in A_i.</math>
  • The Lebesgue integral of a step function <math>f = \sum\limits_{i=0}^n \alpha_i \chi_{A_i}\,</math> is <math>\int \!f\,dx = \sum\limits_{i=0}^n \alpha_i \ell(A_i),\,</math> where <math>\ell(A)</math> is the length of the interval <math>A,</math> and it is assumed here that all intervals <math>A_i</math> have finite length. In fact, this equality (viewed as a definition) can be the first step in constructing the Lebesgue integral.[1]
  • The derivative of a step function is the Dirac delta function
<math>\delta(x) = \begin{cases} +\infty, & x = 0 \\ 0, & x \ne 0 \end{cases}</math>

See also

References

  1. Weir, Alan J. Lebesgue integration and measure. Cambridge University Press, 1973. ISBN 0-521-09751-7. Text "chapter 3" ignored (help)

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