For a real number α, let [α] denote the greatest integer less than or equal to α. For a finite set S, let |S| denote the number of elements in the set S. Consider the functions $f : (-3,3) \to (-$inf, inf) and $g : (-3,3) \to (-$inf, inf) 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 is discontinuous at $x\}$ and $B = \{x$ in (-3,3) : g is discontinuous at $x\}.$ Then the value of |A| + 2|B| - |A ∩ B| is ________.

For a real number α, let [α] denote the greatest integer less than or equal to α. For a finite set S, let |S| denote the number of elements in the set S. Consider the functions f: (-3, 3) → (-∞, ∞) and g: (-3, 3) → (-∞, ∞) defined by f(x) = [x³] logₑ(1 + sin²(π(x - [x]))) and g(x) = x³ sin²(π logₑ(1 + x - [x])). Let A = {x ∈ (-3, 3) : f is discontinuous at x} and B = {x ∈ (-3, 3) : g is discontinuous at x}. Then the value of |A| + 2|B| - |A ∩ B| is ________.

$f(x) = [x^{3}]\log_e\!\left(1+\sin^{2}\!\left(\pi(x-[x])\right)\right)$

$g(x) = x^{3}\sin^{2}\!\left(\pi\log_e(1+x-[x])\right)$

$\text{domain } (-3,3)$

$A = \{x: f \text{ discontinuous}\},\quad B = \{x: g \text{ discontinuous}\}$

$\text{Find } |A| + 2|B| - |A\cap B|$

$1.\ \text{simplify } f$

$2.\ \text{count } A$

$3.\ \text{count } B \text{ and } A\cap B$

$\sin^{2}\!\left(\pi(x-[x])\right) = \sin^{2}(\pi x)$

$\Rightarrow\ f(x) = [x^{3}]\log\!\left(1+\sin^{2}(\pi x)\right)$

$h(x) = \log\!\left(1+\sin^{2}(\pi x)\right) \text{ is continuous}$

$[x^{3}] \text{ jumps where } x^{3}\in\mathbb{Z},\quad x^{3}\in(-27,27)$

$\#\{x^{3}\in\mathbb{Z}\} = 53\quad (n=-26,-25,\ldots,26)$

$\text{jump points: } x = \sqrt[3]{n}$

$h=0 \Leftrightarrow x\in\mathbb{Z}:\ x = -2,-1,0,1,2\ (\text{integer cubes})$

$|A| = 53 - 5 = 48$

$g(x) = x^{3}\sin^{2}\!\left(\pi\log(1+\{x\})\right)$

$x^{3} \text{ continuous} \Rightarrow \text{break only where } \{x\} \text{ jumps (integers)}$

$\text{at integer } n\neq 0:\ n^{3}\sin^{2}(\pi\log 2) \neq g(n)=0\ (\text{jump})$

$\text{at } 0:\ x^{3}=0 \Rightarrow g \text{ continuous}$

$B = \{-2,-1,1,2\} \Rightarrow |B| = 4$

$A\cap B = \varnothing \Rightarrow |A\cap B| = 0$

$|A| + 2|B| - |A\cap B| = 48 + 2(4) - 0 = 56$

$\boxed{56}$