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:

:$$chi_A(x) =left{ egin{matrix} 1, & mathrm{if} ; x in A \ 0, & mathrm{otherwise}. end{matrix} ight.

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