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 $x$.

(N/A) The function is defined as $f: R \rightarrow R$ where $f(x) = [x]$.
To check for one-one:
Consider $f(1.2) = [1.2] = 1$ and $f(1.9) = [1.9] = 1$.
Since $f(1.2) = f(1.9)$ but $1.2 \neq 1.9$,the function is not one-one.
To check for onto:
The range of the greatest integer function is the set of all integers $Z$.
Since the codomain is $R$ and the range $Z \subset R$,there exist elements in the codomain (e.g.,$0.7$) that have no pre-image in the domain.
For example,there is no $x \in R$ such that $f(x) = 0.7$ because $[x]$ must be an integer.
Therefore,the function is not onto.
Hence,the greatest integer function is neither one-one nor onto.