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In this definition, the intervals <math>A_i</math> can be assumed to have the following two properties:
# The intervals are [[disjoint
# The [[union (set theory)|union]] of the intervals is the entire real line: <math>\bigcup_{i=0}^n A_i = \mathbb R.</math>
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* The [[rectangular function]], the normalized [[boxcar function]], is used to model a unit pulse.
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* 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<ref name=bachman_narici_beckenstein>{{Cite book | author=Bachman, Narici, Beckenstein | title=Fourier and Wavelet Analysis | publisher=Springer, New York, 2000 | isbn=0-387-98899-8 | chapter =Example 7.2.2| date=5 April 2002 }}</ref> also define step functions with an infinite number of intervals.<ref name=bachman_narici_beckenstein />
==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 a field|algebra]] over the real numbers.
* A step function takes only a finite number of values. If the intervals <math>A_i,</math> for <math>i=0, 1, \dots, n</math> in the above definition of the step function are disjoint and their union is the real line, then <math>f(x)=\alpha_i</math> for all <math>x\in A_i.</math>
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