Step function

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In mathematics, a function on the real numbers is called a step function (or staircase function) if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces.

File:StepFunctionExample.png
Example of a step function (the red graph). This particular step function is right-continuous.

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 disjoint, for
  • The union 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

File:Dirac distribution CDF.svg
The Heaviside step function is an often used step function.
File:Rectangular function.svg
The rectangular function, the next simplest step function.

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 "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 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.[1]
  • The derivative of a step function is the Dirac delta function

See also

References

  1. Weir, Alan J. Lebesgue integration and measure. Cambridge University Press, 1973. ISBN 0-521-09751-7. Text "chapter 3" ignored (help)

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