January 21, 2022

# Greatest Integer Function

The Greatest integer function $f:\mathbb{R}\rightarrow&space;\mathbb{R}$ is defined as   $f(x)=\left&space;\lfloor&space;x&space;\right&space;\rfloor\,&space;\,&space;for&space;\,&space;\,&space;\,&space;all\,&space;\,&space;x\in&space;\mathbb{R}$,   where  $\left&space;\lfloor&space;\,&space;\,&space;\right&space;\rfloor$  denotes the greatest integer that is less than or equal to $x$.

In set  notation we would write this as   $\left&space;\lfloor&space;x&space;\right&space;\rfloor=max\left&space;\{&space;m\in&space;\mathbb{Z}|m\leq&space;x&space;\right&space;\}$. It is also called the floor function  and step function.

Example.  $\left&space;\lfloor&space;3.7&space;\right&space;\rfloor=3$,    $\left&space;\lfloor&space;-&space;2.3&space;\right&space;\rfloor=-3$  and $\left&space;\lfloor&space;4&space;\right&space;\rfloor=4$

# Domain and Range of Greatest integer function

Domain of greatest integer function is  $\mathbb{R}$ ( the set of all real numbers) and range is $\mathbb{Z}$( the set of all integers ) as it only attains integer value.

## Graph of Greatest integer function

Let $f:\mathbb{R}&space;\rightarrow&space;\mathbb{R}$ be the greatest  integer function define as

$y=\lfloor&space;x&space;\rfloor$   for all $x\in&space;\mathbb{R}$

Let us calculate some value of $y$ for different value of  $x$.

$y=\lfloor&space;x&space;\rfloor&space;=-1,&space;\,&space;\,&space;\,&space;-1\leq&space;x&space;<0$

$y=\lfloor&space;x&space;\rfloor&space;=0,&space;\,&space;\,&space;\,&space;0\leq&space;x&space;<1$

$y=\lfloor&space;x&space;\rfloor&space;=1,&space;\,&space;\,&space;\,&space;1\leq&space;x&space;<2$

$y=\lfloor&space;x&space;\rfloor&space;=2,&space;\,&space;\,&space;\,&space;2\leq&space;x&space;<3$

The graph of greatest integer function is given below.

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

A function $f$ from $A$ to $B$ is said to be one-one if whenever $f(a)=f(b)$ then  $a=b$. It means distinct elements have distinct image.

Here $f:\mathbb{R}&space;\rightarrow&space;\mathbb{R}$ is given by  $f(x)=\left&space;\lfloor&space;x&space;\right&space;\rfloor$,

we can see that    $f(1.3)=\left&space;\lfloor&space;1.3&space;\right&space;\rfloor=1$  and $f(1.7)=\left&space;\lfloor&space;1.7&space;\right&space;\rfloor=1$.

$\therefore&space;f(1.3)=f(1.7),$  but  $1.3\neq&space;1.7$

Hence $f$  is not one-one.

A function $f$ from $A$ to $B$ is called onto if for all $b&space;\in&space;B$ there is an $a\in&space;A$ such that $f(a)=b.$

Now consider $2.7&space;\in&space;\mathbb{R}$.

It is known that  $f(x)=\left&space;\lfloor&space;x&space;\right&space;\rfloor$   is always an integer.  Thus there does not exists any element  $x\in&space;\mathbb{R}$ such that $f(x)=2.7$ .  Thus  $f$ 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 $f(x)$ is said to be continuous at a point c  if each of the following condition is satisfied.

(i) $f(c)$ exists.  (c  is in the domain of $f$)

(ii)  $\lim_{x\rightarrow&space;c}f(x)$ exists and

(iii) $\lim_{x\rightarrow&space;c}f(x)=f(c)$

Geometrically, this means that there is no gap, split  or missing point for $f(x)$ at  $c$ and that a pencil could be moved along the graph of f(x)  through $(c,&space;f(c))$ without lifting it off the graph.

Greatest integer function is continous at all points apart from integral values of $x$. It is also differentiable with ${f}'(x)=0$ , at all points apart from integral values of $x$.

Let $c$ be an integer .Then we can find a sufficiently small  real number $r>0$ such that $\left&space;\lfloor&space;c-r&space;\right&space;\rfloor=c-1$  whereas $\left&space;\lfloor&space;c+r&space;\right&space;\rfloor=c$. Thus

$\lim_{x\rightarrow&space;c^{-}}f(x)=\lim_{x\rightarrow&space;c^{-}}\left&space;\lfloor&space;x&space;\right&space;\rfloor=c-1$  and $\lim_{x\rightarrow&space;c^{+}}f(x)=\lim_{x\rightarrow&space;c^{+}}\left&space;\lfloor&space;x&space;\right&space;\rfloor=c$.

Since  these limits cannot be equal  to each other for any $c$ , 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 $\lim_{x\rightarrow&space;1.5^{-}}\left&space;\lfloor&space;x&space;\right&space;\rfloor=\lim_{x\rightarrow&space;1.5^{+}}\left&space;\lfloor&space;x&space;\right&space;\rfloor=1=&space;\left&space;\lfloor&space;1.5\right&space;\rfloor$

In general,   if  $c$ is any  number which is not an integer and  $n-1$n$ is an integer , then    $\lim_{x\rightarrow&space;c}\left&space;\lfloor&space;x&space;\right&space;\rfloor&space;=n-1=\left&space;\lfloor&space;c&space;\right&space;\rfloor$.

#### Bina singh

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