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Step Function: Definition, Equation & Examples Video

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  • 0:00 Step Function
  • 1:33 Greatest Integer Function
  • 2:10 Least Integer Function
  • 3:17 Heaviside Function
  • 4:00 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we'll learn what a step function is and become comfortable with this type of function. We'll examine common examples, such as the floor and ceiling functions and the heaviside function.

Step Function

Mathematically speaking, a step function is a function whose graph looks like a series of steps because it consists of a series of horizontal line segments with jumps in-between. For this reason, it is also sometimes called a staircase function.

A step function has a constant value on given intervals, but the constant is different for each interval. The constant value on each interval creates the series of horizontal lines, and the fact that the constant is different for each interval creates the jumps in between each horizontal line segment. This is why the graph of a step function looks like a set of stairs.

To help us better understand this definition, let's consider a real-world example. Suppose I start a cleaning business. I decide to charge each client based on how many hours I work for that client.

My charge structure is as follows:

  • Less than one hour: $10.00
  • One hour up to two hours: $20.00
  • Two hours up to three hours: $30.00
  • Three hours up to four hours: $40.00

This means that if I work anywhere from 1-59 minutes, I charge $10.00. If I work anywhere from 1 hour to 1 hour and 59 minutes, I charge $20.00, and so on. We see that the amount I charge remains constant on each hour-long interval, but varies on each of those intervals, because it goes up each hour. Here is the graph corresponding to this example.

cleaning business graph

It is easy to see from this graph why a step function is sometimes called a staircase function; it looks exactly like a set of stairs.

Greatest Integer Function

In mathematics, a common example used to introduce step functions is the greatest integer function (also called the floor function). The greatest integer function is often represented as x with bottom brackets around it.

greatest integer function

It maps each real number x to the greatest integer that is less than or equal to x. Whatever we put into the greatest integer function, we get the greatest integer that is less than or equal to that input as our output. Here is the graph of the greatest integer function.

greatest integer function

It is easy to see that the greatest integer function is a step function from its graph.

Least Integer Function

Let's consider another example of a step function that is very similar to the greatest integer function. It is called the least integer function (also known as the ceiling function). The least integer function is often represented as x with top brackets around it.

Least Integer Function

The least integer function is a step function that assigns, or maps, each real number x to the smallest integer that is greater than or equal to x. Whatever we put into the least integer function, we get the smallest integer that is less than or equal to that input as our output. The least integer function is shown in the graph on screen.

least integer function

We see that the graph of the least integer function looks like a set of stairs, as it should since this is a step function.

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