Step function

This article is about a piecewise constant function. For the unit step function, see Heaviside step function.

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.

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 (sometimes written as ) is the indicator function of :

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

  1. The intervals are pairwise disjoint, for
  2. 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

The Heaviside step function is an often-used step function.
The rectangular function, the next simplest step function.

Non-examples

Properties

See also

References

  1. 1 2 Bachman, Narici, Beckenstein. "Example 7.2.2". Fourier and Wavelet Analysis. Springer, New York, 2000. ISBN 0-387-98899-8.
  2. Weir, Alan J. "3". Lebesgue integration and measure. Cambridge University Press, 1973. ISBN 0-521-09751-7.
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