Consider the function $f : \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \to (-\infty, \infty)$ defined by $f(x) = (|x| + |x - 1|) \sin x + [x \sin x]$, where $[x \sin x]$ is the greatest integer less than or equal to $x \sin x$. Let $\alpha$ be the total number of points in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ at which $f$ is NOT continuous, and let $\beta$ be the total number of points in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ at which $f$ is NOT differentiable. Then the value of $\alpha + \beta$ is ________.

$f(x) = (|x| + |x-1|)\sin x + [\,x\sin x\,],\quad x \in \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$ $\text{To find: } \alpha + \beta$ $\text{Plan: simplify } |x|+|x-1|;\ \text{jumps of } [x\sin x];\ \text{check kinks}$ $|x|+|x-1| = \begin{cases} 1-2x, & -\tfrac{\pi}{2}<x<0\\ 1, & 0\le x<1\\ 2x-1, & 1\le x<\tfrac{\pi}{2}\end{cases}$ $\text{Only } [\,x\sin x\,] \text{ jumps: } h(x)=x\sin x \in \left[0,\dfrac{\pi}{2}\right)$ $[h] \text{ jumps where } h \text{ is an integer}$ $h(0)=0,\qquad h\left(\pm\tfrac{\pi}{2}\right)=\tfrac{\pi}{2}\approx 1.57$ $\Rightarrow \text{integers crossed: } 0 \text{ and } 1$ $h=0 \text{ only at } x=0:\ [h]=0 \text{ both sides} \Rightarrow \text{no jump}$ $h=1 \text{ at } x=\pm x_1\ (\approx \pm 1.11) \Rightarrow \text{jump} \Rightarrow \alpha = 2$ $\text{Non-diff. points} \supseteq \{\pm x_1\} \cup \{\text{kinks of } |x|,\ |x-1|\}$ $x=0:\quad f'^- = \dfrac{d}{dx}[(1-2x)\sin x] = 1,\quad f'^+ = \dfrac{d}{dx}[\sin x] = 1$ $\Rightarrow f \text{ is differentiable at } x=0$ $x=1:\quad f'^- = \cos 1,\quad f'^+ = 2\sin 1 + \cos 1$ $2\sin 1 + \cos 1 \neq \cos 1 \Rightarrow \text{not differentiable at } x=1$ $\beta = 2\ (\text{at } \pm x_1) + 1\ (\text{at } x=1) = 3$ $\alpha + \beta = 2 + 3 = 5$ $\boxed{5}$