Prove that the Greatest Integer Function f: R → R, given by f(x) … - Vidyadip

Question

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

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Answer

f: R → 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 $1.2\neq1.9$ $\therefore$ f is not one-one. Now, consider $0.7\in\text{R}.$
It is known that f(x) = [x] is always an integer. Thus, there does not exist any element $\text{x}\in\text{R}$ 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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