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 $$f: \mathbb{R} \rightarrow \mathbb{R}$$ 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 $$\chi_A\,$$ is the indicator function of $$A$$:


 * $$\chi_A(x) =

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

In this definition, the intervals $$A_i$$ can be assumed to have the following two properties:


 * The intervals are disjoint, $$A_i\cap A_j=\emptyset$$ for $$i\ne j$$


 * 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


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

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 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 "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 $$x\in A_i.$$
 * The Lebesgue integral of a step function $$f = \sum\limits_{i=0}^n \alpha_i \chi_{A_i}\,$$ is $$\int \!f\,dx = \sum\limits_{i=0}^n \alpha_i \ell(A_i),\,$$ 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.
 * The derivative of a step function is the Dirac delta function


 * $$\delta(x) = \begin{cases} +\infty, & x = 0 \\ 0, & x \ne 0 \end{cases}$$