Prove that the greatest integer function $f:R\to R$, given $f(x)=\left[ x \right]$ is neither one-one nor onto where $\left[ x \right]$ denotes the greatest integer function less than or equal to $x$.

Hint: In this question we have to prove that the greatest integer function which is denoted as $f(x)=\left[ x \right]$ is neither a one-one function nor an onto function. We will first see the definition of a one-one and onto function and then prove both the cases by taking examples in the range of $f:R\to R$.

Complete step by step answer:
The Greatest Integer Function is denoted by $f(x)=\left[ x \right]$ which means it is less than or equal to $x$. it rounds a real number to the nearest or the closest integer.
A function is supposed to be one-one when there are no two elements in the domain of $f(x)$ which correspond to the same element which is in the range of $f(x)$.
Now consider the example of the greatest integer function of $1.2$, It can be written mathematically as:
$f(1.2)=\left[ 1.2 \right]$
Since the greatest integer which is not greater than $1.2$ is not an integer and $1$ is the greatest integer not greater than $1$, we can write the solution as:
 $f(1.2)=1$
Now consider the example of the greatest integer function of $1.9$, It can be written mathematically as:
$f(1.9)=\left[ 1.9 \right]$
Since the greatest integer which is not greater than $1.9$ is not an integer and $1$ is the greatest integer not greater than $1$, we can write the solution as:
 $f(1.9)=1$
Now from the two examples we can see that both the functions point to the same value in the range of the function $f(x)$ therefore, we can conclude that it is not a one-one function.
Now a function being onto means that all the elements in the range of $f(x)$ have at least one solution in the domain of the function.
Now the domain for the function given is $f:R\to R$ which is the real number.
Since the greatest integer function always returns an integer value and no real value, we can conclude that it is not an onto function either.

Note: The greatest integer function is called as the floor function since it returns the greatest integer from the real number. There also exists the least integer function which is also called as the ceiling function which gives the least integer which is greater than or equal to $x$. The floor function is denoted as $f(x)=\left\lceil x \right\rceil $ and the ceiling function is called $f(x)=\left\lfloor x \right\rfloor $.