Question

A function $f$ defined as $f\left(x\right)=x\left[x\right]$ for $-1\le x\le 3$ where $\left[x\right]$ defines the greatest integer $\le x$ is-

• A. continuous at all points in the domain of $f$ but non-derivable at a finite number of points
• B. discontinuous at all points and hence non-derivable at all points in the domain of $f$
• C. discontinuous at a finite number of points but not derivable at all points in the domain of $1$
• D. discontinuous and also non-derivable at a finite number of points of $f$

Answer 1

Answer: (D)

discontinuous and also non-derivable at a finite number of points of $f$

We can write the given function as $f\left(x\right)=\{\begin{array}{c}-x,-1\le x<0\\ 0,0\le x<1\\ x,1\le x<2\\ 2x,2\le x<3\\ 9,x=3\end{array}$
Thus, the function is discontinuous at discrete points, namely $x=0,1,2,3.$
Since it is discontinuous, its also not differentiable.