Question

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

Answer 1

$f\left(x\right)=\left[x\right]$

It is seen that $f\left(1.2\right)=\left[1.2\right]=1,f\left(1.9\right)=\left[1.9\right]=1$.

$\therefore f\left(1.2\right)=f\left(1.9\right)$, but $1.2\ne 1.9$
$\therefore f$ is not one-one
Now, consider $0.7\in R$
It is known that $f\left(x\right)=\left[x\right]$ is always an integer. Thus, there does not exist any element $x\in R$ such that $f\left(x\right)=0.7$
$\therefore f$ is not onto.
Hence, the greatest integer function is neither one-one nor onto.