Greatest Integer Function
The greatest integer function is also known as the step function. The greatest integer function rounds up the number to the nearest integer less than or equal to the given number. The greatest integer function has a step curve which we will explore in the following sections. The domain of the greatest integer function is \(\mathbb{R}\) and its range is \(\mathbb{Z}\).
Therefore the greatest integer function is simply rounding off to the greatest integer that is less than or equal to the given number. Here we shall learn more about the greatest integer function, its graph, and the properties.
What is Greatest Integer Function?
Greatest Integer Function is a function that gives the greatest integer less than or equal to the number. The greatest integer less than or equal to a number x is represented as ⌊x⌋. We will round off the given number to the nearest integer that is less than or equal to the number itself. Clearly, the input variable x can take on any real value. However, the output will always be an integer. Also, all integers will occur in the output set.
Thus, the domain of this function is real numbers (\(\mathbb{R}\)), while its range will be integers (\(\mathbb{Z}\)). Look at the following examples of the greatest integer function in the following table:
Values of x  f(x)=⌊x⌋ 

3.1  f(3.1) = ⌊3.1⌋ = 3 
2.999  f(2.999) = ⌊2.999⌋ = 2 
−2.7  f(−2.7) = ⌊−2.7⌋ = −3 
−√2  f(−√2) = ⌊−√2⌋ = −2 
4  f(4) = ⌊4⌋ = 4 
−7  f(−7) = ⌊−7⌋ = −7 
Greatest Integer Function Graph
The greatest integer function graph is known as the step curve because of the step structure of the curve. Let us plot the greatest integer function graph. First, consider f(x) = ⌊x⌋, if x is an integer, then the value of f will be x itself. If x is a noninteger, then the value of x will be the integer just before x.
For example,
 For all numbers lying in the interval [0,1), the value of f will be 0.
 For the entire interval [1,2), f will take the value 1.
 For the interval [−1,0), f will take the value −1 and so on.
So for an integer n, [n, n+1) will have the value of the greatest integer function as n. The function has a constant value between any two integers. As soon as the next integer comes, the function value jumps by one unit. This means that the value of f at x = 1 is 1 (and not 0) hence there will be a hollow dot at (1,0) and a solid dot at (1,1) where a hollow dot means not including the value and solid dot represents including the value. These observations lead us to the following graph.
Domain and Range of Greatest Integer Function
From the graph above we can clearly see that inputs of the function can be any real number but the output will always be the integers. Thus, the domain of this function is real numbers (\(\mathbb{R}\)), while its range will be integers (\(\mathbb{Z}\)).
Greatest Integer Function Properties
There are various properties related to greatest integrer function. Some useful greatest integer function properties are listed below.
 ⌊x+n⌋ = ⌊x⌋+n, where, \(n \in \mathbb{Z}\)
 ⌊−x⌋ \(\begin{cases} & {\left\lfloor x\right\rfloor}, & \text{if} x \in \mathbb{Z} \\ &{\left\lfloor x1\right\rfloor}, & \text{if} x \notin \mathbb{Z} \\ \end{cases}\)
 If ⌊f(x)⌋ ≥ L, then f(x)≥L
 If x is a number between successive integers n and n+1, then ⌊x⌋=n. If x is an integer, then ⌊x⌋=x
 The domain of the greatest integer function is \(\mathbb{R}\) and its range is \(\mathbb{Z}\).
 The fractional part will always be nonnegative, as x will always be greater than (or equal to) ⌊x⌋. If x is an integer, then its fractional part will be 0
 The domain of the fractional part function is \(\mathbb{R}\) and its range is [0,1).
Related Topics
The following links are related to the greatest integer function
Greatest Integer Function Examples

Example 1:
What is the domain of the given greatest integer functions :
a) f(x)=1/⌊x⌋
(b) g(x)=1{x} ?
Solution:
(a) The denominator should not be 0, that is, ⌊x⌋≠0
The greatest integer part of a number is 0 if that number lies in the interval [0,1)
Thus, to obtain the domain, this interval must be excluded from the set of real numbers.
This means that the domain of \(f\) is \(\mathbb{R} \left[ {0,1} \right)\)
(b) Once again, the denominator should not be 0, which means that {x}≠0
When is the fractional part of a number equal to 0?
Only when that number is an integer!
Thus, the domain of g is \(\mathbb{R}\mathbb{Z}\), that is, all integers have been excluded from the set of real numbers.
Answer: The domain of \(f\) is \(\mathbb{R} \left[ {0,1} \right)\) and the domain of \(g\) is \(\mathbb{R}\mathbb{Z}\).

Example 2: Find the value of x such that ⌊x+3⌋ = 3
Solution:
From the definition of the greatest integer function, we have 3 ≤ x+1 < 4
Subtract 1 in this inequality.
We get, 2 ≤ x < 3
Answer: x can take the values greater than or equal to 2 and less than 3.
FAQs on Greatest Integer Function
Where is Greatest Integer Function Not Differentiable?
As we check the graph of the greatest integer function, we can see that it is jumping whenever it reaches an integer. Since the curve is discontinuous at integers, it is not differentiable at those points. Therefore at each integer, the greatest integer function is not differentiable.
What is Greatest Integer Function?
The greatest integer function is a function that gives the largest integer which is less than or equal to x. This function is denoted by ⌊x⌋. We will round off the given number to the nearest integer that is less than or equal to the number itself.
Is the Floor Function Differentiable?
The floor function or the greatest integer function is not differentiable at integers. The floor function has jumping values at integers, so its curve is known as the step curve. The curve of floor function is discontinuous at integers and hence not differentiable at integers.
What is the Domain and Range of the Greatest Integer Function?
The input of the greatest integer function can be any real number whereas the greatest integer function's output is always an integer. Also, all integers will occur in the output set. Thus, the domain of this function is real numbers (\(\mathbb{R}\)), while its range will be integers (\(\mathbb{Z}\)).
How to Graph Greatest Integer Function?
Plotting the graph of the greatest integer function is easy. It is a step curve. Here, f(x) = ⌊x⌋, if x is an integer, then the value of f will be x itself and if x is a noninteger, then the value of x will be the integer just before x. Hence for an integer n, [n, n+1) will have the value of the greatest integer function as n. The function has a constant value between any two integers. As soon as the next integer comes, the function value jumps by one unit. This means that the value of f at x = 1 is 1 (and not 0) hence there will be a hollow dot at (1,0) and a solid dot at (1,1) where a hollow dot means not including the value and solid dot represents including the value. And similarly, we can plot the curve of the greatest integer function.