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Step function

 

 
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Step function
 
 
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In mathematicsMathematics

Mathematics is the discipline that deals with concepts such as quantity, structure, space and change....
, a functionFunction (mathematics)

In mathematics, a function relates each of its inputs to exactly one output....
 on the real numberReal number

In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers....
s is called a step function (or staircase function) if it can be written as a finiteFinite set

In mathematics, a set is called finite if there is a bijection between the set and some set of the form where is a nat...
 linear combinationLinear combination

In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics....
 of indicator functionIndicator function

In mathematics, an indicator function or a characteristic function is a function defined on a set that indicates memb...
s of intervalInterval (mathematics)

In elementary algebra, an interval is a set that contains every real number between two indicated numbers and possibly the t...
s. Informally speaking, a step function is a piecewisePiecewise

In mathematics, a function f of a real number variable x is defined piecewise, if it is continuous on all but a fini...
 constant functionConstant function

In mathematics a constant function is a function whose values do not vary and thus are constant....
 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 functionIndicator function

In mathematics, an indicator function or a characteristic function is a function defined on a set that indicates memb...
 of :

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

  • The intervals are disjoint, for


  • The unionUnion (set theory)

    In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that ...
     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

  • A constant functionConstant function

    In mathematics a constant function is a function whose values do not vary and thus are constant....
     is a trivial example of a step function. Then there is only one interval,
  • The Heaviside functionHeaviside step function

    The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a disc...
     H(x) is am important step function. It is the mathematical concept behind some test signalFacts About Signal

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

    In control theory the unit step response is the response of a dynamic system to the Heaviside step function....
     of a dynamical systemDynamical system (definition)

    The dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a p...
    .
  • 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 algebraAlgebra over a field Summary

    In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a ...
     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.

See also

  • Simple functionSimple function

    In mathematics, especially in mathematical analysis, a simple function is a measurable function whose range is finite....
  • Piecewise defined function
  • Sigmoid functionSigmoid function

    A sigmoid function is a mathematical function that produces a sigmoid curve — a curve having an "S" shape....