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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 on the real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s is called a step function (or staircase function) if it can be written as a finite
Finite set

In mathematics, finite set is a Set that has a finite number of element . For example,is a finite set with five elements. The number of elements of a finite set is a natural number , and is called the cardinality of the set....
 linear combination
Linear combination

In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics.Most of this article deals with linear combinations in the context of a vector space over a field , with some generalisations given at the end of the article....
 of indicator function
Indicator function

In mathematics, an indicator function or a characteristic function is a Function defined on a Set that indicates membership of an element in a subset of ....
s of interval
Interval (mathematics)

In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set....
s. Informally speaking, a step function is a piecewise
Piecewise

In mathematics, a piecewise-defined function is a function whose definition is dependent on the value of the independent variable. Mathematically, a real number-valued function f of a real variable x is a relationship whose definition is given differently on disjoint subsets of its domain ....
 constant function
Constant function

In mathematics, a constant function is a function whose values do not vary and thus are constant. For example, if we have the function f = 4, then f is constant since f maps any value to 4....
 having only finitely many pieces.
nction 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
Indicator function

In mathematics, an indicator function or a characteristic function is a Function defined on a Set that indicates membership of an element in a subset of ....
 of :

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





Indeed, if that is not the case to start with, a different set of intervals can be picked for which these assumptions hold.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 on the real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
s is called a step function (or staircase function) if it can be written as a finite
Finite set

In mathematics, finite set is a Set that has a finite number of element . For example,is a finite set with five elements. The number of elements of a finite set is a natural number , and is called the cardinality of the set....
 linear combination
Linear combination

In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics.Most of this article deals with linear combinations in the context of a vector space over a field , with some generalisations given at the end of the article....
 of indicator function
Indicator function

In mathematics, an indicator function or a characteristic function is a Function defined on a Set that indicates membership of an element in a subset of ....
s of interval
Interval (mathematics)

In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set....
s. Informally speaking, a step function is a piecewise
Piecewise

In mathematics, a piecewise-defined function is a function whose definition is dependent on the value of the independent variable. Mathematically, a real number-valued function f of a real variable x is a relationship whose definition is given differently on disjoint subsets of its domain ....
 constant function
Constant function

In mathematics, a constant function is a function whose values do not vary and thus are constant. For example, if we have the function f = 4, then f is constant since f maps any value to 4....
 having only finitely many pieces.
Stepfunctionexample

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
Indicator function

In mathematics, an indicator function or a characteristic function is a Function defined on a Set that indicates membership of an element in a subset of ....
 of :

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

  • The intervals are disjoint, for


  • The union
    Union (set theory)

    In set theory, the term Union refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets....
     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



Examples

Dirac Distribution Cdf
* A constant function
Constant function

In mathematics, a constant function is a function whose values do not vary and thus are constant. For example, if we have the function f = 4, then f is constant since f maps any value to 4....
 is a trivial example of a step function. Then there is only one interval,
  • The Heaviside function
    Heaviside step function

    The Heaviside step function, H, also called the unit step function, is a continuous function Function whose value is 0 for negative argument and 1 for positive argument....
     H(x) is an important step function. It is the mathematical concept behind some test signal
    Signal

    Signal, signals, signaling, or signalling may refer to:...
    s, such as those used to determine the step response
    Step response

    The step response of a system in a given initial state consists of the time evolution of its outputs when its input are Heaviside step functions....
     of a dynamical system
    Dynamical system (definition)

    The dynamical system concept is a mathematics formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space....
    .
  • 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
    Algebra over a field

    In mathematics, an algebra over a field is an algebraic structure consisting of a vector space together with an Binary operation, usually called multiplication, that combines any two vectors to form a third vector....
     over the real numbers.
  • A step function takes only a finite number of values. If the intervals in the above definition of the step function are disjoint and their union is the real line, then for all
  • 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.
  • The derivative
    Derivative

    In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
     of a step function is the Dirac delta function
    Dirac delta function

    The Dirac delta or Dirac's delta is a mathematics construct introduced by theoretical physicist Paul Dirac. Informally, it is a function representing an infinitely sharp peak bounding unit area: a function d that has the value 0 everywhere except at x = 0 where its value is infinity in such a way that its total integral is 1....




See also

  • Simple function
    Simple function

    In mathematics field of real analysis, a simple function is a real number-valued function over a subset of the real line which attains only a finite number of values....
  • Piecewise defined function
  • Sigmoid function
    Sigmoid function

    Many natural processes and complex system learning curve display a history dependent progression from small beginnings that accelerates and approaches a climax over time....