Step function

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

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

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:

The intervals are disjoint, $\scriptstyle A_i \,\cap\, A_j ~=~ \emptyset$ for $\scriptstyle i ~\ne~ j$
The union of the intervals is the entire real line, $\scriptstyle \cup_{i=0}^n A_i ~=~ \mathbb R.$

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

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, $A_0=\mathbb R.$
The Heaviside function H(x) is an important step function. 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.

The rectangular function, the normalized boxcar function, is the next simplest step function, and is used to model a unit pulse.

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 intervals. However, some authors define step functions also with an infinite number of intervals.^[1]

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 $A_i,$ $i=0, 1, \dots, n,$ 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]

References

↑ for example see: Bachman, Narici, Beckenstein. "Example 7.2.2". Fourier and Wavelet Analysis. Springer, New York, 2000. ISBN 0-387-98899-8.
↑ Weir, Alan J. "3". Lebesgue integration and measure. Cambridge University Press, 1973. ISBN 0-521-09751-7.

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