Continuity and Differenti question bank (Maths)

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 $f : (-3, 3) \rightarrow (-\infty, \infty)$ and $g : (-3, 3) \rightarrow (-\infty, \infty)$ defined by
$f(x) = [x^3]\log_e(1 + \sin^2(\pi(x - [x])))$
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 ___________.

Consider the function $f : \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right) \rightarrow (-\infty, \infty)$ 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 $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$ at which $f$ is NOT continuous, and let $\beta$ be the total number of points in the interval $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$ at which $f$ is NOT differentiable.

Then the value of $\alpha + \beta$ is ___________.