Question

Let the functions $f:(-1,1)\to R$ and $g:(-1,1)\to (-1,1)$ be defined by $f\left(x\right)=|2x-1|+|2x+1|$ and $g\left(x\right)=x-\left[x\right],$ where $\left[x\right]$ denotes the greatest integer less than or equal to $x.$ Let $f\circ g:(-1,1)\to R$ be the composite function defined by $(f\circ g)\left(x\right)=f\left(g\left(x\right)\right).$ Suppose $c$ is the number of points in the interval $(-1,1)$ at which $f\circ g$ is NOT continuous, and suppose $d$ is the number of points in the interval $(-1,1)$ at which $f\circ g$ is NOT differentiable.Then the value of $c+d$ is

Answer 1

$f\left(x\right)=|2x-1|+|2x-1|$
$f\left(x\right)=\{\begin{array}{cc}-4x,& x\le \frac{-1}{2}\\ 2,& \frac{-1}{2}<x<\frac{1}{2}\\ 4x,& x\ge \frac{1}{2}\end{array}$

$g\left(x\right)=x-\left[x\right]=\text{{}x\text{}}$

Now, $f\circ g\left(x\right)=\{\begin{array}{cc}-4g\left(x\right),& g\left(x\right)\le \frac{-1}{2}\\ 2,& \frac{-1}{2}<g\left(x\right)<\frac{1}{2}\\ 4g\left(x\right),& g\left(x\right)\ge \frac{1}{2}\end{array}\Rightarrow f\circ g\left(x\right)=\{\begin{array}{cc}2,& x\in [0,1/2)\cup (-1,-1/2)\\ 4x,& x\in [1/2,1)\\ 4(1+x),& x\in [-1/2,0)\end{array}$


$f\circ g$ is not continuous at $x=0$ only.
$\Rightarrow c=1$
$f\circ g$ is not differentiable at $x=\frac{-1}{2},0,\frac{1}{2}$
$\Rightarrow d=3$
$c+d=4$