Let R be the set of real number and `f:R to R` be defined by `f(x)=({x})/(1+[x]^(2))`, where [x] is the greatest integer less than or equal to x, and {x}=x-[x]. Which of the following statement are true?
I. The range of f is a closed interval
II. f is continuous on R.
III. f is one - one on R.

Options

A. I only
B. II only
C. III only
D. None of I, II and III

Answer

D

Explanation

`f{x}=({x})/(1+[x]^(2))`
`f(x)={{:((x+1)/(2):-1lex lt0),(x:0lexlt1),((x-1)/(2),1lexlt2),((x-2)/(2),2lexlt3),(" "underset("So-on")(vdots)):}`
Now check accordingly