GREATEST INTEGER FUNCTION

Definition Of Greatest Integer Function

The greatest integer function of a real number x is represented by f(x) = [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

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Video Examples: The Greatest Integer Function
 

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, the value of the function f(x) remains – 5 in the interval [–5, –4)

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

Ques :

(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