Step function

Pronunciation: /stɛp ˈfʌŋk ʃən/ ?
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=(-infinity,-2]; B=(-2,-1); C=[-1,2]; D=(2,4]; E=(4,infinity)
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.

More Information

  • Piecewise Function. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem.org. 2009-11-19. http://www.allmathwords.org/article.aspx?lang=en&id=Piecewise%20Function.
  • Ceiling Function. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem.org. 2009-11-19. http://www.allmathwords.org/article.aspx?lang=en&id=Ceiling%20Function.
  • Floor Function. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem.org. 2009-11-19. http://www.allmathwords.org/article.aspx?lang=en&id=Floor%20Function.
  • Step Functions. bookrags.com. Macmillan Science Library: Mathematics. Macmillan Reference USA. 2009-11-19. http://www.bookrags.com/research/step-functions-mmat-04/.

Cite this article as:
Step function. 2009-11-19. All Math Words Encyclopedia. Life is a Story Problem.org. http://www.allmathwords.org/en/s/stepfunction.html.

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2009-11-19: Initial version. (McAdams, David.)

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