**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. For example, the value of the greatest integer function is**n****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*_|. - Greatest integer function is also called floor function.

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

**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

- Step Function
- Interval
- Integer
- Piece-wise defined function
- Domain
- Floor function