Step function

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This article is about a piecewise constant function. For the unit step function, see Heaviside step function.

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 f: \mathbb{R} \rightarrow \mathbb{R} is called a step function if it can be written as[citation needed]

f(x) = \sum\limits_{i=0}^n \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\, (sometimes written as 1_A) is the indicator function of A:

\chi_A(x) =
\begin{cases}
1 & \mbox{if } x \in A, \\
0 & \mbox{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 disjoint, A_i\cap A_j=\emptyset for i\ne j
  2. The union of the intervals is the entire real line, \cup_{i=0}^n A_i=\mathbb 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, 1)} + 3 \chi_{(0, 6)}\,

can be written as

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

Examples[edit]

The Heaviside step function is an often used step function.
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 define step functions also with an infinite number of intervals.[1]

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 A_i, i=0, 1, \dots, 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 Lebesgue integral of a step function \textstyle f = \sum\limits_{i=0}^n \alpha_i \chi_{A_i}\, is \textstyle \int \!f\,dx = \sum\limits_{i=0}^n \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.[2]

See also[edit]

References[edit]

  1. ^ for example see: 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.