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.

Definition and first consequences

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

:$f(x)\; =\; sumlimits\_\{i=0\}^n\; alpha\_i\; chi\_\{A\_i\}(x),$ for all real numbers $x$

where $nge\; 0,$ $alpha\_i$ are real numbers, $A\_i$ are intervals, and $chi\_A,$ is the indicator function of $A$:

:

In this definition, the intervals $A\_i$ can be assumed have the following two properties:

* The intervals are disjoint, $A\_icap\; A\_j=emptyset$ for $i\; e\; j$

* The union of the intervals is the entire real line, $cup\_\{i=1\}^n\; A\_i=mathbb\; R.$

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

: $f\; =\; 0chi\_\{(-infty,\; -5)\}\; +4\; chi\_\{\; [-5,\; 0]\; \}\; +7\; chi\_\{(0,\; 1)\}\; +\; 3\; chi\_\{\; [1,\; 6)\}+0chi\_\{\; [6,\; infty)\}.,$

Examples

* 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 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 $A\_i,$ $i=0,\; 1,\; dots,\; n,$ in the above definition of the step function are disjoint and their union is the real line, then $f(x)=alpha\_i,$ for all $xin\; A\_i.$
* The Lebesgue integral of a step function $f\; =\; sumlimits\_\{i=0\}^n\; alpha\_i\; chi\_\{A\_i\},$ is $int\; !f,dx\; =\; sumlimits\_\{i=0\}^n\; alpha\_i\; ell(A\_i),,$ 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. [Cite book | author=Weir, Alan J | authorlink= | coauthors= | title=Lebesgue integration and measure | date= | publisher=Cambridge University Press, 1973 | location= | isbn=0-521-09751-7 | chapter 3]

ee also

*Simple function
*Piecewise defined function
*Sigmoid function

References