Question

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

Answer 1

Here, $f:R\to R$, 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 $1.2\ne 1.9$
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.