December 8, 2022

Greatest Integer Function

The Greatest integer function f:\mathbb{R}\rightarrow \mathbb{R}

is defined as   f(x)=\left \lfloor x \right \rfloor\, \, for \, \, \, all\, \, x\in \mathbb{R},   where  \left \lfloor \, \, \right \rfloor  denotes the greatest integer that is less than or equal to x.

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

Example.  \left \lfloor 3.7 \right \rfloor=3,    \left \lfloor - 2.3 \right \rfloor=-3  and \left \lfloor 4 \right \rfloor=4

greatest integer functionDomain 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} \rightarrow \mathbb{R} be the greatest  integer function define as

y=\lfloor x \rfloor   for all x\in \mathbb{R}

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

y=\lfloor x \rfloor =-1, \, \, \, -1\leq x <0

y=\lfloor x \rfloor =0, \, \, \, 0\leq x <1

y=\lfloor x \rfloor =1, \, \, \, 1\leq x <2

y=\lfloor x \rfloor =2, \, \, \, 2\leq x <3

The graph of greatest integer function is given below.

graph of greatest integer function

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} \rightarrow \mathbb{R} is given by  f(x)=\left \lfloor x \right \rfloor,

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

\therefore f(1.3)=f(1.7),  but  1.3\neq 1.7

Hence f  is not one-one.

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

Now consider 2.7 \in \mathbb{R}.

It is known that  f(x)=\left \lfloor x \right \rfloor   is always an integer.  Thus there does not exists any element  x\in \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 c}f(x) exists and

(iii) \lim_{x\rightarrow 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, 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 \lfloor c-r \right \rfloor=c-1  whereas \left \lfloor c+r \right \rfloor=c. Thus

\lim_{x\rightarrow c^{-}}f(x)=\lim_{x\rightarrow c^{-}}\left \lfloor x \right \rfloor=c-1  and \lim_{x\rightarrow c^{+}}f(x)=\lim_{x\rightarrow c^{+}}\left \lfloor x \right \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 1.5^{-}}\left \lfloor x \right \rfloor=\lim_{x\rightarrow 1.5^{+}}\left \lfloor x \right \rfloor=1= \left \lfloor 1.5\right \rfloor

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









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