Step Function
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In mathematics, a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s is called a step function if it can be written as a
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
linear combination 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 , one has \mathbf_(x)=1 if x\i ...
s of intervals. Informally speaking, a step function is a
piecewise In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ...
constant function In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image). Basic properties ...
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 , one has \mathbf_(x)=1 if x\i ...
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 mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A c ...
: A_i \cap A_j = \emptyset for i \neq j # The
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...
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 In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image). Basic properties ...
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 The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
, 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 In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The ''IEEE Transactions on Signal Processing'' ...
, such as those used to determine the
step response The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the out ...
of a
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
. * The
rectangular function The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as \operatorname(t) = \Pi(t) = \left\{\begin{array}{rl ...
, the normalized boxcar function, is used to model a unit pulse.


Non-examples

* The
integer part In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least int ...
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 one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
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 In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
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 A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
is sometimes defined as a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
whose
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...
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 In mathematics, the crenel function is a periodic discontinuous function ''P''(''x'') defined as 1 for ''x'' belonging to a given interval and 0 outside of it. It can be presented as a difference between two Heaviside step function The Heavis ...
* Piecewise defined function *
Sigmoid function A sigmoid function is a mathematical function having a characteristic "S"-shaped curve or sigmoid curve. A common example of a sigmoid function is the logistic function shown in the first figure and defined by the formula: :S(x) = \frac = \ ...
*
Simple function In the mathematical field of real analysis, a simple function is a real (or complex)-valued function over a subset of the real line, similar to a step function. Simple functions are sufficiently "nice" that using them makes mathematical reasoning, ...
*
Step detection In statistics and signal processing, step detection (also known as step smoothing, step filtering, shift detection, jump detection or edge detection) is the process of finding abrupt changes (steps, jumps, shifts) in the mean level of a time seri ...
*
Unit step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
*
Piecewise-constant valuation A piecewise-constant valuation is a kind of a function that represents the utility of an agent over a continuous resource, such as land. It occurs when the resource can be partitioned into a finite number of regions, and in each region, the value-d ...


References

{{DEFAULTSORT:Step Function Special functions