KVPY SX SB Exam 05-Nov-2017 Question Paper

Let R be the set of real numbers and f : R → R be defined by $\mathrm{f}\left(\mathit{x}\right)=\frac{\left\{\mathit{x}\right\}}{1+{\left[\mathit{x}\right]}^2}$ , where [x] is the greatest integer less than or equal to x, and {x} = x – [x]. Which of the following statements are true ?
I. The range of f is a closed interval
II. f is continuous on R.
III. f is one-one on R.