Step Functions

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.

Frederick Landwehr

Bibliography

Miller, Charles D., Vern E. Heeren, and E. John Hornsby, Jr. Mathematical Ideas, 9th ed. Boston: Addison-Wesley, 2001.