question_answer 19)

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.  

Answer:

 given by f(x) = [x]       Injectivity : Let x1 = 2.5 and x2 = 2 be two elementsof R.       f(x1) = f(2.5) = [2.5] = 2       f(x2) = f(2) = [2] = 2        f(x1) = f(x2) for        is not one-one.       Surjectivity : Let y = 2.5  be any element        f(x) = 2.5             which is not possible as [x] is always an integer.        is not onto.