The Greatest integer function is defined as

In set notation we would write this as . It is also called the floor function and step function.

Example.

**Domain and Range of Greatest integer function**

Domain of greatest integer function is

**Graph of Greatest integer function**

Let

for all

Let us calculate some value of for different value of

The graph of greatest integer function is given below.

**Greatest integer function is neither one-one nor onto**

A function

Here is given by

we can see that and

but

Hence is not one-one.

A function

Now consider

It is known that is always an integer. Thus there does not exists any element

Hence , the Greatest integer function is neither one-one nor onto.

**Continuity and differentiability of greatest integer function**

Continuity and differentiability are properties of a function at a specific point rather than properties of a function as a whole . A function is said to be continuous at a point **c **if each of the following condition is satisfied.

(i)

(ii)

(iii)

Geometrically, this means that there is no gap, split or missing point for

Greatest integer function is continous at all points apart from integral values of . It is also differentiable with

Let

Since these limits cannot be equal to each other for any

The greatest integer function is continuous at evety real no. other than integers. For example

Let’s take x=1.5 . Then

In general, if