Step Function

Pronunciation: /stɛp ˈfʌŋk.ʃən/ Explain
Graph showing the piecewise function f(x)={-1,x<=-2; 1,-2<x<-1; 2,-1<=x<=2; 0,2<x<=4; 1,x>4;}
Figure 1: Step function.

A step function is a piecewise function that is constant over a finite number of intervals. A step function can also be called a staircase function.

A step function is piecewise. Over each of the 'pieces', each interval over the domain, the value of the function remains constant. Figure 1 shows a step function with five intervals labeled A, B, C, D and E as follows:
A = (-∞, -2]
B = (-2, -1)
C = [-1, 2]
D = (2, 4]
E = (4, -∞)
Over each of those intervals, the value of f(x) is constant. For example, over interval C, f(x) = 2.

Examples of Step Functions

Graph showing the ceiling function. A solid point at (-2,-1) is connected by a horizontal line to a hollow point at (-1,-1). A solid point at (-1,0) is connected by a horizontal line to a hollow point at (0,0). A solid point at (0,1) is connected by a horizontal line to a hollow point at (1,1). This pattern repeats infinitely in both directions. A graph showing the floor function.
Figure 2: Ceiling function. Figure 3: Floor function.

Examples of functions that not step functions

A function that is not a step function.

This function is not a step function because it is not constant on the interval [-1, 1].

 A function that is not a step function.
Figure 4: Not a step function. Figure 5: Not a step function.

References

  1. McAdams, David E.. All Math Words Dictionary, step function. 2nd Classroom edition 20150108-4799968. pg 171. Life is a Story Problem LLC. January 8, 2015. Buy the book

More Information

  • Piecewise Function. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 11/19/2009. https://www.allmathwords.org/en/p/piecewisefunction.html.

Cite this article as:

 McAdams, David E. Step Function. 5/7/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/s/stepfunction.html.

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Revision History

5/7/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
12/10/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
8/7/2018: Changed vocabulary links to WORDLINK format. (McAdams, David E.)
7/18/2018: Changed title to common format. (McAdams, David E.)
11/19/2009: Initial version. (McAdams, David E.)

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