Prove that the Greatest Integer Function $f: R \rightarrow R ,$ given by $f(x)=[x]$, is neither one-one nor onto, where $[x]$ denotes the greatest integer less than or equal to $x$.
Prove that the Greatest Integer Function $f: R \rightarrow R ,$ given by $f(x)=[x]$, is neither one-one nor onto, where $[x]$ denotes the greatest integer less than or equal to $x$.
$f : R \rightarrow R$ is given by, $f ( x )=[ x ]$
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 $0.7 \in R$
It is known that $f(x)=[x]$ is always an integer. Thus, there does not exist any element $x \in R$ such that $f(x)=0.7$
$\therefore f$ is not onto
Hence, the greatest integer function is neither one-one nor onto.
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