Question

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

Answer 1

$f(x)=|2 x-1|+|2 x+1|$
$g(x)=\{x\}$
$f(g(x))=\begin{cases} 2 & -1 < x < -\frac{1}{2} \\ 4 x+4 & -\frac{1}{2} \leq x < 0 \\ 2 & 0 \leq x < \frac{1}{2} \\ 4 x & \frac{1}{2} \leq x < 1\end{cases}$
$\Rightarrow $ Dis-continuous at $x=0 \Rightarrow c=1$
$\Rightarrow $ non-differentiable at $x=-\frac{1}{2}, 0, \frac{1}{2}$
$ \Rightarrow d=3$
$\Rightarrow c+d=4$