In mathematics, functions describe relationships between two or more quantities. A step function is a special type of relationship in which one quantity increases in steps in relation to another quantity.
For example, postage cost increases as the weight of a letter or package increases. In the year 2001 a letter weighing between 0 and 1 ounce required a 34-cent stamp. When the weight of the letter increased above 1 ounce and up to 2 ounces, the postage amount increased to 55 cents, a step increase.
A graph of a step function f gives a visual picture to the term "step function." A step function exhibits a graph with steps similar to a ladder.
The domain of a step function f is divided or partitioned into a number of intervals. In each interval, a step function f (x ) is constant. So within an interval, the value of the step function does not change. In different intervals, however, a step function f can take different constant values.
One common type of step function is the greatest-integer function. The domain of the greatest-integer function f is the real number set that is divided into intervals of the form …[ −2, −1), [−1, 0), [0, 1), [1, 2), [2, 3),… The intervals of the greatest-integer function are of the form [k, k + 1), where k is an integer. It is constant on every interval and equal to k.
f(x) = 0 on [0, 1), or 0≤x <1
f(x) = 1 on [1, 2), or 1≤x <2
f(x) = 2 on [2, 3), or 2≤x <3
For instance, in the interval [2, 3), or 2≤x <3, the value of the function is 2. By definition of the function, on each interval, the function equals the greatest integer less than or equal to all the numbers in the interval. Zero, 1, and 2 are all integers that are less than or equal to the numbers in the interval [2, 3), but the greatest integer is 2.
Therefore, in general, when the interval is of the form [k, k + 1), where k is an integer, the function value of greatest-integer function is k. So in the interval [5, 6), the function value is 5. The graph of the greatest integer function is similar to the graph shown below.
There are many examples where step functions apply to real-world situations. The price of items that are sold by weight can be presented as a cost per ounce (or pound) graphed against the weight. The average selling price of a corporation's stock can also be presented as a step function with a time period for the domain.
Miller, Charles D., Vern E. Heeren, and E. John Hornsby, Jr. Mathematical Ideas, 9th ed. Boston: Addison-Wesley, 2001.
"Step Functions." Mathematics. . Encyclopedia.com. (April 17, 2019). https://www.encyclopedia.com/education/news-wires-white-papers-and-books/step-functions
"Step Functions." Mathematics. . Retrieved April 17, 2019 from Encyclopedia.com: https://www.encyclopedia.com/education/news-wires-white-papers-and-books/step-functions
Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).
Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.
Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the most-recent information available at these sites:
Modern Language Association
The Chicago Manual of Style
American Psychological Association
- Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
- In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.