Greatest Integer Function

When the intervals are in the form of (n, n+1), the value of greatest integer function is n, where n is an integer.

For example, the greatest integer function of the interval [3,4) will be 3.

The graph is not continuous. For instance, below is the graph of the function f(x) = ⌊ x ⌋.

The above graph is viewed as a group of steps and hence the integer function is also called a Step function. The left endpoint in every step is blocked(dark dot) to show that the point is a member of the graph, and the other right endpoint (open circle) indicates the points that are not part of the graph.

You can observe that in every interval, the function f(x) is the same. The function’s value stays constant within an interval. For instance, the value of function f(x) is equal to -5 in the interval [-5, -4).

Let’s Workout: 

Example 1: Find the greatest integer function for following

(a)  ⌊-261⌋

(b)  ⌊3.501⌋

(c) ⌊-1.898⌋

Solution:

According to the greatest integer function definition

(a) ⌊-261⌋ = -261

(b) ⌊3.501⌋ = 3

(c) ⌊-1.898⌋ = -2

Example 2:  Evaluate ⌊3.7⌋.

Solution: 

On a number line, ⌊3.7⌋ lies between 3 and 4

The largest integer which is less than 3.7 is 3.

So, ⌊3.7⌋ = 3 Answer!

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