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 numbers 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
Traditionally, a finite verb (from la, fīnītus, past particip ...
linear combination of
indicator functions 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. ...
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 properti ...
having only finitely many pieces.
Definition and first consequences
A function
is called a step function if it can be written as
:
, 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
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 ...
:
for
# 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
** ''U ...
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
:
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 properti ...
is a trivial example of a step function. Then there is only one interval,
* The
sign function
In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as . To a ...
, 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 argumen ...
, 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
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 of a
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
.
* The
rectangular function, the normalized
boxcar function
In mathematics, a boxcar function is any function which is zero over the entire
real line except for a single interval where it is equal to a constant, ''A''. The boxcar function can be expressed in terms of the uniform distribution as
\operato ...
, 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 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
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
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 Le ...
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.
* A
discrete random variable is sometimes defined as a
random variable 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.
Ever ...
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 functions of amplitude ...
*
Piecewise defined function
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 ...
*
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 = \f ...
*
Simple function
*
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
*
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