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

$f(x) = |2x-1| + |2x+1|, \qquad g(x) = x - [x]$

$[x] = \text{greatest integer } \le x, \qquad x \in (-1,1)$

$c = \#\{\text{points where } f\circ g \text{ not continuous}\}$

$d = \#\{\text{points where } f\circ g \text{ not differentiable}\}$

$\text{Find } c + d$

$1.\ \text{write } g \text{ piecewise}$

$2.\ \text{write } f \text{ on the range of } g$

$3.\ \text{compose and find the break points}$

g(x) = x+1, -1 < x < 0 x, 0 ≤ x < 1

$\Rightarrow g\big((-1,1)\big) = (0,1)$

f(t) = 2, 0 < t < 1/2 4t, 1/2 ≤ t < 1 (f∘ g)(x) = 2, -1<x<-1/2 4(x+1), -1/2≤ x<0 2, 0≤ x<1/2 4x, 1/2≤ x<1

$x\to 0^-: \ f\circ g \to 4(0+1) = 4; \quad x\to 0^+: \ f\circ g \to 2$

$f\circ g(0) = f(0) = 2$

$\Rightarrow c = 1$

$x = -\tfrac12: \ 0 \to 4 \ (\text{corner})$

$x = \tfrac12: \ 0 \to 4 \ (\text{corner})$

$x = 0: \ \text{not continuous} \Rightarrow \text{not differentiable}$

$\Rightarrow d = 3$

$c + d = 1 + 3 = 4$