Greatest integer function returns the greatest integer less than or equal to a real number. In other words, we can say that greatest integer function rounds “down” any number to the nearest integer. This function is also known by the names of “floor” or “step” function. The greatest integer function (GIF) is denoted by the symbol “[x]” .

Interpretation of Greatest integer function is straight forward for positive number. Consider the values “0.23” and “1.7”. The greatest integers for two numbers are “0” and “1”. Now, consider a negative number “-0.54” and “-2.34”. The greatest integers less than these negative numbers are “-1” and “-3” respectively.

We can observe here that greater integer function is actually a function that returns the integral part of a positive real number. This interpretation is clear for positive number. Interpretation for negative numbers needs some explanation. We interpret these values in the context of the fact that every real number can be decomposed to have two parts (i) integral and (ii) fractional part. From this point of view, the negative number can be thought as :

We may be tempted to disagree (why not -2 + -0.34 = -2.34?). But, we should know that this is how greatest integer function (GIF) treats a negative number. It returns "-3" for "-2.34" - not "-2". Subsequently, we shall define a function called fraction part function (FPF) that returns fraction part of real number. We shall find that the function exactly returns the same fraction for negative number as has been worked out. The fraction part function (FPF) returns a fraction, which is always positive. It is denoted as {x}. Because of these aspects of GIF and FPF, we can understand the reason why negative number is treated the way it has been presented above. In terms of integral and fraction parts, we write a real number "x" as :

In the nutshell, we can use any of the following interpretations of greatest integer function :

- [x] = Greatest integer less than equal to “x”
- [x] = Greatest integer not greater than “x”
- [x] = Integral part of “x”

The value of "[x]" is an integer (n) such that :

Working rules for evaluating greatest integer function are two step process :

- If “x” is an integer, then [x] = x.
- If “x” is not an integer, then [x] evaluates to greatest integer less than “x”.

**Graph of greatest integer function **

Few initial function values are :

The graph of the function is shown here :

Greatest integer function |
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This function is known as step function as values of function steps by "1" as we switch values of “x” from one interval to another. We see that there is no restriction on values of "x" and as such its domain has the interval equal to that of real numbers. On the other hand, the step function or greatest integer function evaluates only to integer values. It means that the range of the function is set of integers, denoted by "Z". Hence,

GIF is not a periodic function. Though function is defined for all real x, but graph is not continous. It breaks at integral values of x.

#### Example 1

Problem : Find domain of function given by :

Solution : The denominator of function is positive. This means :

The value of π is 3.14. Here, [x] returns integral value. Clearly, it can assume a maximum value of 3. But, GIF returns integer value “n” for x<n+1. The inequality, therefore, has solution given by :

**Important properties **

Certain properties of greatest integer function are presented here :

1: If and only if “x”is an integer, then :

2: If and only if at least either “x” or “y” is an integer, then :

For example, let x = -2.27 and y = 0.63. Then,

However, if one of two numbers is integer like x = -2 and y = 0.63, then the proposed identity as above is true.

4: If “x” belongs to integer set, then :

For example, let x = 2.Then

We can use this identity to test whether “x” is an integer or not?

3: If “x” does not belong to integer set, then :

For example, let x = 2.7.Then

#### Example 2

Problem : Find domain of the function :

Solution : Given function is in rational form having GIF as its denominator. The denominator should not evaluate to zero for real values of x. The domain of GIF is real number set R. But, we know that GIF evaluates to zero in an interval which is spread over unit value. In order to know this interval, we determine interval of x for which [x-2] is zero.

We can write this function as :

Using property [x+y] = [x] + [y], provided one of x and y is an integer. This is the case here,

Hence, domain of given function is :