Question

Let $f\left(x\right)=|\left[x\right]x|$ for $-1\le x\le 2$ , where $\left[x\right]$ denotes greatest integer function, then

• A. $f\left(x\right)$ is discontinuous at $x=0$
• B. $f\left(x\right)$ is differentiable at $x=1$
• C. $f\left(x\right)$ is not differentiable at $x=2$
• D. $f\left(x\right)$ is differentiable at $x=2$

Answer 1

Answer: (C)

$f\left(x\right)$ is not differentiable at $x=2$

We divide the interval in 4 parts,
For $-1<x<0$
$f\left(x\right)=-x$
For $0<x<1$
$f\left(x\right)=0$
For $1<x<2$
$f\left(x\right)=x$
For $x=2$
$f\left(x\right)=4$

Now at $x=0$,
lim x $\to 0^{-}$ f(x) = 0 = lim x $\to 0^{+}$ f(x)
Hence, f(x) is continuous at x = 0
At x = 1
lim x $\to 1^{-}$ f(x) = 0 not equal to lim x $\to 1^{+}$ f(x)
Hence, f(x) is discontinuous at x = 1
At x = 2
lim x $\to 2^{-}$ f(x) = 0 not equal to lim x = 2 f(x)
Hence, f(x) is discontinuous as well as non-differentiable at $x=2$