December 8, 2023

# Greatest Integer Function

The Greatest integer function is defined as

,   where    denotes the greatest integer that is less than or equal to
.

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

Example.

,      and

# Domain and Range of Greatest integer function

Domain of greatest integer function is

( the set of all real numbers) and range is ( the set of all integers ) as it only attains integer value.

## Graph of Greatest integer function

Let

be the greatest  integer function define as

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

from to
is said to be one-one if whenever then
. It means distinct elements have distinct image.

Here is given by

,

we can see that      and

.

but

Hence   is not one-one.

A function

from to
is called onto if for all there is an
such that

Now consider

.

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

such that .  Thus
is not onto.

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)

exists.  (c  is in the domain of )

(ii)

exists and

(iii)

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

at  and that a pencil could be moved along the graph of f(x)  through
without lifting it off the graph.

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

, at all points apart from integral values of .

Let

be an integer .Then we can find a sufficiently small  real number such that
whereas . Thus

and .

Since  these limits cannot be equal  to each other for any

, the function is discontinous at every integral points.

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

is any  number which is not an integer and
is an integer , then    . #### Bina singh

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