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Linear combination of indicator functions of real intervals From Wikipedia, the free encyclopedia

In mathematics, a function on the real numbers is called a **step 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.

A function is called a **step function** if it can be written as ^{[citation needed]}

- , for all real numbers

where , are real numbers, are intervals, and is the indicator function of :

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

- The intervals are pairwise disjoint: for
- The union of the intervals is the entire real line:

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

can be written as

Sometimes, the intervals are required to be right-open^{[1]} or allowed to be singleton.^{[2]} The condition that the collection of intervals must be finite is often dropped, especially in school mathematics,^{[3]}^{[4]}^{[5]} though it must still be locally finite, resulting in the definition of piecewise constant functions.

- A constant function is a trivial example of a step function. Then there is only one interval,
- The sign function sgn(
*x*), which is −1 for negative numbers and +1 for positive numbers, and is the simplest non-constant step function. - The Heaviside function
*H*(*x*), which is 0 for negative numbers and 1 for positive numbers, is equivalent to the sign function, up to a shift and scale of range (). 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 used to model a unit pulse.

- 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
^{[6]}also define step functions with an infinite number of intervals.^{[6]}

- 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 for in the above definition of the step function are disjoint and their union is the real line, then for all
- The definite integral of a step function is a piecewise linear function.
- The Lebesgue integral of a step function is where is the length of the interval , and it is assumed here that all intervals have finite length. In fact, this equality (viewed as a definition) can be the first step in constructing the Lebesgue integral.
^{[7]} - A discrete random variable is sometimes defined as a random variable whose cumulative distribution function is piecewise constant.
^{[8]}In this case, it is locally a step function (globally, it may have an infinite number of steps). Usually however, any random variable with only countably many possible values is called a discrete random variable, in this case their cumulative distribution function is not necessarily locally a step function, as infinitely many intervals can accumulate in a finite region.

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