Algebra of Continuous Functions

Consider the function $$f:\left(-\tfrac{\pi}{2},\tfrac{\pi}{2}\right)\to(-\infty,\infty)$$ 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 $$\left(-\tfrac{\pi}{2},\tfrac{\pi}{2}\right)$$ at which $$f$$ is NOT continuous, and let $$\beta$$ be the total number of points in the interval $$\left(-\tfrac{\pi}{2},\tfrac{\pi}{2}\right)$$ at which $$f$$ is NOT differentiable. Then the value of $$\alpha+\beta$$ is ___.

For a real number $$\alpha$$, let $$[\alpha]$$ denote the greatest integer less than or equal to $$\alpha$$. For a finite set $$S$$, let $$|S|$$ denote the number of elements in the set $$S$$.

Consider the functions $$f:(-3,3)\to(-\infty,\,\infty)$$ and $$g:(-3,3)\to(-\infty,\,\infty)$$ defined by

$$f(x)=[x^3]\log_e\big(1+\sin^2(\pi(x-[x])))\big)$$

and

$$g(x)=x^3\sin^2(\pi\log_e(1+x-[x])).$$

Let

$$A=\{x\in(-3,3):f\text{ is discontinuous at }x\}$$

and

$$B=\{x\in(-3,3):g\text{ is discontinuous at }x\}.$$

Then the value of $$|A|+2|B|-|A\cap B|$$ is ___.