Question

Let $f\left(x\right)=\left[x\right],g\left(x\right)=|x|$ and $f\left(g\right(x\left)\right)=h\left(x\right)$, where [.] is the greatest integer function. Then $h(-1)$ is

• A. $0$
• B. $-\infty$
• C. non existent
• D. none of these

Answer 1

Answer: (C)

non existent

$f\left(x\right)=\left[x\right]$ and $g\left(x\right)=|x|$
$\Rightarrow f\left(g\right(x\left)\right)=\left[|x|\right]$
$h\left(x\right)=\left[|x|\right]$
$h(-1^{+})=0$
$h(-1)=1$
$h(-1^{-})=1$
Clearly, the RHS, LHS limits do not match at $x=-1$, hence it is not differentiable