Definition of Greatest Integer Function

• The greatest integer function of a real number x is represented by [x] or |_x_|.

• For all real numbers x, the greatest integer function returns the largest integer less than or equal to x.


In other words, the greatest integer function rounds down a real number to the nearest integer.

More about Greatest Integer Function

• Greatest integer functions are piece-wise defined.



• The domain of the greatest integer function is the set of real numbers which is divided into a number of intervals like [4, 3), [3, 2), [2, 1), [1, 0), [0, 1), [1, 2), [2, 3), [3, 4) and so on.


Hint: [a, b) is just an interval notation which means a ≤ x < b, where x is a real number in the interval
        [a, b).
 
When the interval is of the form [n, n + 1), where n is an integer, the value of the greatest integer function is n. For example, the value of the greatest integer function is 4 in the interval [4, 3).


• The graph of a greatest integer function is not continuous. For example, the following is the graph of the greatest integer function f (x) = |_x_|.



The graph above looks like a stair case (a series of steps). So, the greatest integer function is sometimes called a step function. One endpoint in each step is closed (black dot) to indicate that the point is a part of the graph and the other endpoint is open (open circle) to indicate that the points is Not a part of the graph.

Observe in the graph above that in each interval, the function f(x) is constant. Within an interval, the value of the function remains constant. For example, in the interval [–5, –4) the value of the function f(x) remains – 5.




In different intervals, however, the function f(x) can take different constant values.


• Greatest integer function is also called floor function.

Solved Example on Greatest Integer Function

Find:

(a) |_-256_|

(b) |_3.506_|

 

(c) |_-0.7_|

Solution:

 

By the definition of greatest integer function,

(a) |_-256_| = -256

(b) |_3.506_| = 3

(c) |_-0.7_| = -1

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