The Greatest Integer Function - Problem 1

I want to graph the greatest integer function. I want to take a slightly different approach because I could plot a lot of points. But greatest integer function behaves differently from other functions that we’ve studied. Let’s start with the idea of what the greatest integer function is.

Remember it’s the greatest integer less than or equal to x, the input. For example if I have .5 here, the greatest integer less than or equal to .5 is 0. Let’s think about all the numbers whose greatest integer is 0. 0 works, .5 works, .9 works, .99, and every number between 0 and 1 but not including 1, will have a greatest integer of 0. What that means is, when we graph this function; it’s going to look like this. It’s a little open circle here at 1, because the greatest integer of 1 is not 0. But all this other points will have a value of 0.

Now what happens at 1? The greatest integer of 1 is 1. The greatest integer of 1.1 is also 1. And the greatest integer of any number between 1 and 2, is going to be 1 unless it’s actually 2. Any number from 1 to 2, not including 2. So here, all these numbers from 1 to 2, but not 2, they have a greatest integer of 1.

What about 2? The greatest integer of 2 is 2. The greatest integer of any number larger than 2 but less than 3 is 2 but the greatest integer of 3 is 3. So we have another one of these little half-open segments. The graph is going to continue to look like this. It’ll continue to be these little segments closed on one end, open on the other; closed on the left, open on the right.

Is it the same for negative values? It is. Think about what numbers have a greatest integer of -1. Other than -1, the numbers over here; greater than -1 but less than 0. So an output of -1. All of these numbers from -1 to 0 but not including. So these things do just continue down the negative direction. We’ve got another little segment half open. Instead of trying to draw all of them, I’ll just put a little ellipsis at both ends to show the pattern continues.

So remember this graph. It’s really quite unique in the sense that it’s composed of these segments each one unit long, closed on the left end, open on the right. Also, notice what that does to the domain and range? The domain, every real number has the greatest integer less than or equal to it. So the domain is all real numbers. But notice that the range is only the integers. And the symbol for the integers is the bold face Z. So the range is the integers; the domain is the set of real numbers.