Prove that the Greatest Integer Function


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


$f: \mathbf{R} \rightarrow \mathbf{R}$ is given by,


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

$\therefore f(1.2)=f(1.9)$, but $1.2 \neq 1.9$.

$\therefore f$ is not one-one.

Now, consider $0.7 \in \mathbf{R}$.

It is known that $f(x)=[x]$ is always an integer. Thus, there does not exist any element $x \in \mathbf{R}$ such that $f(x)=0.7$.

∴ f is not onto.

Hence, the greatest integer function is neither one-one nor onto.

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