Question

Let R be the set of real numbers and $f:R\to R$ be defined by $f\left(x\right)=\frac{\left\{x\right\}}{1+[x{]}^2}$, where $\left[x\right]$ is the greatest integer less than or equal to x, and $\left\{x\right\}=x-\left[x\right]$. 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.

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

Answer 1

The correct option is D None of I, II and III

At every integer value of $x,f\left(x\right)=0$

If $0<x<1\Rightarrow f\left(x\right)=\left\{x\right\}$

If $2<x<3\Rightarrow f\left(x\right)=\frac{\left\{x\right\}}{2}$ and

If $-1<x<0\Rightarrow f\left(x\right)$ is not defined$\Rightarrow$ range of $f\left(x\right)$ is not closed interval.

If $-2<x<-1\Rightarrow f\left(x\right)=-\frac{\left\{x\right\}}{1}$ and so on..

It is evident from the intervals above that the values of the function above that every value of f(x) does not have exactly one element in the range.

$\therefore f\left(x\right)$ is not continuous and one one function on $R$.

Thus the correct answer is $D$