Question

Let the functions $f:(-1,1)\to R$ and $g:(-1,1)\to (-1,1)$ be defined by
$f(x)=|2x-1|+|2x+1|$ and $g(x)=x-[x]\text{, }$
where $[x]$ denotes the greatest integer less than or equal to $x$. Let fog : $(-1,1)\to R$ be the composite function defined by $(fog)(x)=f(g(x))$. 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

Answer: 4

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