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In mathematics, a function on the
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s is called a step function if it can be written as a finite
linear combination In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...
s of intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces.


Definition and first consequences

A function f\colon \mathbb \rightarrow \mathbb is called a step function if it can be written as :f(x) = \sum\limits_^n \alpha_i \chi_(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 In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...
of A: :\chi_A(x) = \begin 1 & \text x \in A \\ 0 & \text x \notin A \\ \end In this definition, the intervals A_i can be assumed to have the following two properties: # The intervals are
pairwise disjoint In set theory in mathematics and Logic#Formal logic, formal logic, two Set (mathematics), sets are said to be disjoint sets if they have no element (mathematics), element in common. Equivalently, two disjoint sets are sets whose intersection (se ...
: A_i \cap A_j = \emptyset for i \neq j # The union of the intervals is the entire real line: \bigcup_^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_ + 3 \chi_ can be written as :f = 0\chi_ +4 \chi_ +7 \chi_ + 3 \chi_+0\chi_.


Variations in the definition

Sometimes, the intervals are required to be right-open or allowed to be singleton. The condition that the collection of intervals must be finite is often dropped, especially in school mathematics, though it must still be locally finite, resulting in the definition of piecewise constant functions.


Examples

* A constant function is a trivial example of a step function. Then there is only one interval, A_0=\mathbb R. * The sign function , which is −1 for negative numbers and +1 for positive numbers, and is the simplest non-constant step function. * The Heaviside function , 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 (H = (\sgn + 1)/2). 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.


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 also define step functions with an infinite number of intervals.


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 Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
over the real numbers. * A step function takes only a finite number of values. If the intervals A_i, for 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 definite integral of a step function is a piecewise linear function. * The Lebesgue integral of a step function \textstyle f = \sum_^n \alpha_i \chi_ is \textstyle \int f\,dx = \sum_^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. * A discrete random variable is sometimes defined as a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
whose cumulative distribution function is piecewise constant. 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.


See also

* Crenel function * Piecewise *
Sigmoid function A sigmoid function is any mathematical function whose graph of a function, graph has a characteristic S-shaped or sigmoid curve. A common example of a sigmoid function is the logistic function, which is defined by the formula :\sigma(x ...
* Simple function * Step detection * Heaviside step function * Piecewise-constant valuation


References

{{DEFAULTSORT:Step Function Special functions