If $f(x)=|x-1|+|x-2|+|x-3|$ for $2 < x < 3$,then $f$ is

Given that for any $n \in N$,there exist an odd integer $q$ and a non-negative integer $r$ such that $n$ can be written uniquely as $n = q \times 2^r$. Let $f: N \rightarrow N \times N$ be a function defined by $f(n) = \left(r+1, \frac{q+1}{2}\right)$. Then,

If a function $f: R \rightarrow R$ is defined by $f(x)=x^3-x$,then $f$ is

The mapping $f: R \to R$ defined as $f(x) = \cos x, x \in R$ is:

If $R$ denotes the set of all real numbers,then the function $f: R \to R$ defined by $f(x) = [x]$ is:

Let $R$ be the set of real numbers and $f: R \rightarrow R$ be defined by $f(x) = \frac{\{x\}}{1+[x]^2}$,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 $R$.