# CLASS-11RELATION & FUNCTION - THE GREATEST INTEGER FUNCTION

THE GREATEST INTEGER FUNCTION (OR STEP FUNCTION) -

The greatest integer function, often denoted as ⌊x⌋, returns the largest integer less than or equal to a given real number x. In other words, it rounds down to the nearest integer. Here's its formal definition:

⌊x⌋ = the greatest integer less than or equal to x

For example:

• ⌊3.7⌋ = 3 ⌊3.7⌋ = 3
• ⌊−2.5⌋ = −3 ⌊−2.5⌋ = −3

It's important to note that the greatest integer function only returns integers. If you're working with rational or real numbers, the result will still be an integer. This function is often used in mathematical modeling, especially in contexts where you need to discretize continuous variables or when you want to find the largest integer value that fits within a certain constraint.

Example.1) Draw the graph of the function y = {x} in -2 < x < 4

Ans.) y = [x]. Here the value of y is the integer value of x. Some points of the graphs are -

- 2, when x ∈ [-2, -1)

- 1, when x ∈ [- 1, 0)

f(x) = [        0, when x ∈ [0, 1)

1, when x ∈ [1, 2)

and so on

The required graph is shown in above figure. Clearly the function has jumps at the points (-1, -2), (0, -1), (1, 0), (2, 1), etc. In other words, the given function is discontinuous at each internal value of x.