Proving Points of Continuity for Greatest Integer Function

What is the greatest integer function? The greatest integer function rounds up the number to the most neighboring integer less than or equal to the provided number. This function has a step curve and is also recognized as the step function. The domain and range of the greatest integer function is R and Z respectively. It is also known as the floor of X. Where [x] denotes the largest integer that is less than or equal to x.

Hence the greatest integer function is clearly rounding off to the greatest integer that is less than or equivalent to the provided number. Herewith this article we shall learn about the greatest integer function domain and range with graphs, properties, and examples.

Check out this article on Relations and Functions.

What is the Greatest Integer Function?

As read in the introduction, the greatest integer function is a function that presents the greatest integer which is less than or equivalent to the number. Such a number that is less than or equal to a number x is depicted by the notation ⌊x⌋.

\(\text{ In general }:If,n\le X<n+1.\ \text{Then } ,(n\epsilon \text{ Integer })\Longrightarrow[X]=n\)

This implies if X lies in \([n,\ n+1)\), then the Greatest Integer Function of X will be n.

In this method, we will simply round off the assigned number to the most adjacent integer that is smaller than or equal to the number itself. Obviously, the input variable x can have any real value. But, the output will always be an integer. Some of the examples of the greatest integer function are given in the tabular format:

Values of x	f(x)=⌊x⌋
3.2	f(3.2) = ⌊3.2⌋ = 3
2.888	f(2.888) = ⌊2.888⌋ = 2
−√3	f(−√3) = ⌊−√3⌋ = −3
−7	f(−7) = ⌊−7⌋ = −7
−2.88	f(−2.88) = ⌊−2.88⌋ = −3
6	f(6) = ⌊6⌋ = 6

Also, read about Linear Inequality here.

Greatest Integer Functions Examples

If the table above is not that clear as to how you can determine the greatest integer function of a given number then go through these examples:

Example 1: ⌊2.4⌋

Remember that number we are looking for must satisfy two conditions.

• The number should be an integer one.
• The number should be lesser than or equal to 2.4.

So a number that is smaller than 2.4 and is an integer is 2.

Therefore ⌊2.4⌋ = 2

Example 2: ⌊-2.66⌋

Again, the number we are viewing for must satisfy the following two conditions.

• The number should be an integer one.
• The number should be lesser than or equal to -2.66.
• A common mistake that students mostly commit is to assume that [-2.66] = -2.
• For ⌊2.66⌋ = 2, it resembles that we just lifted the .5. But that is not the case with negative numbers and ⌊-2.66⌋ = -2 is the wrong answer.

⌊-2.66⌋ is not equal to -2.

Recollect that we are searching for a number smaller than -2.66. Hence the number that is smaller than -2.66 is -3.

So ⌊-2.66⌋= -3

Example 3: ⌊5⌋

Here we are viewing for a number lesser than or equal to 5. As 5 is equivalent to 5.

Therefore ⌊5⌋= 5

Example 4: ⌊0. 56⌋

Till now it would be certain that we would focus on the number that is less or equal and try to neglect as much as possible the word greatest. So the integer that is less than 0.56 is 0.

Since it is 0, ⌊0. 56⌋ = 0.

Check more topics of Mathematics here.

Domain and Range of Greatest Integer Functions

The range of the greatest integer function is an integer that is (Z) and the domain of the greatest integer function is R i.e any real number. This implies that for any graph the inputs of the function can take any real number but the output will constantly be an integer.

That is a function represented by [x] recited as step ′x′

It is specified for all x where the domain=(−∞,∞) and the range is all integers.

Greatest Integer Function Graph

The greatest integer function graph is also identified as the step curve because of the step formation of the curve. Let's understand the graph of the greatest integer function through a plot. Suppose f(x) = ⌊x⌋, if x is an integer, then the value of f will be x itself and if x is not an integer, then the value of x will be the integer just smaller than x.

For example,

For all numbers resting in the interval [0,1), the output of f will be 0. That is:

For all numbers resting in the interval [−1,0), f will use the value −1 and so on for the next set of numbers.

x	-1	-0.5	-0.4	-0.2	0
f(x)	-1	-1	-1	-1	0

Similarly, for all numbers resting in the entire interval [1,2), f will take the value 1.

• So for an integer m, [m, m+1) will hold the value of the greatest integer function as m. From the graph, we can say that the function has a fixed value within any two integers.
• As soon as the subsequent integer appears, the function value shifts by one unit. This indicates that the value of f at x = 1 is 1.
• Therefore there will be a hollow dot at the location (1,0) and a solid dot at the location (1,1).
• Wherein the hollow dot means not involving the value and solid dot signifies including the value. These observations direct us to the above graph.

If you are reading Greatest Integer Functions can also go through Limit and Continuity.

Properties of Greatest Integer Functions

There are different properties associated with the greatest integer function some of them are as follows:

\(⌊x+n⌋=⌊x⌋+n,\text{ where },\ n\ ∈Z\)

\(⌊x⌋=x\ \text{ exists if x is an integer }\ .\)

\(⌊−x⌋=−⌊x⌋,\ if\ x\ ∈Z\text{ (If x is an Integer)}\)

\(⌊−x⌋=−⌊x⌋−1,\ ifx\ ∉\ Z\text{ (If x is not an Integer)}\)

\(\text{ If }⌊\ f(x)⌋\ge L,\text{ then }f(x)\ge L\)

Learn more about Logarithmic functions here.

Greatest Integer Functions Key Takeaways

Here is a summary of the gif function:

• If x is a number that lies between successive integers m and m+1, then ⌊x⌋=m.
•  If and only if x is an integer, then the value of ⌊x⌋=x.
• The domain of the greatest integer function is R(all real values) and its range is Z(set of integers).
• The fractional part of a given number will constantly be non-negative, as x will always be higher than or equal to ⌊x⌋. If x is an integer, then its fractional component will be zero.
• The domain of the fractional part function exists in R and its range is [0,1).

We hope that the above article on Greatest Integer Functions is helpful for your understanding and exam preparations. Stay tuned to the Testbook app for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.

Greatest Integer Functions FAQs

Q.1 What is the range of the greatest integer function?

Ans.1 The range of the greatest integer function is an integer that is (Z).

Q.2 What is the greatest integer function of 1?

Ans.2 The greatest integer function of 1 is 1.

Q.3 What is the domain of the greatest integer function?

Ans.3 The domain of the greatest integer function is R i.e any real number.

Q.4 What is the greatest integer?

Ans.4 The greatest integer function is a function that presents the greatest integer which is less than or equivalent to the number. Such a number that is less than or equal to a number say x and is depicted by the notation ⌊x⌋.

Q.5 What is the greatest integer function of 0?

Ans.5 The greatest integer function of 0 is zero.

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