8. Let A = \(\mathbb{R}\setminus\{2, 0, \frac{-\pi}{6}, \frac{-5\pi}{6}, \frac{-7\pi}{6}, ...\}\) and B = \(\mathbb{R}\setminus\{\frac{-\pi}{6}, \frac{-5\pi}{6}, ...\}\). Consider the function $f: A \to \mathbb{R}$ given by $f(x) = \begin{cases} \frac{x^2 - 2x}{x^2 - x - 2} & x > 0, x \neq 2, \\ \tan(\pi \sin x) & x < 0, x \neq \frac{-\pi}{6}, \frac{-5\pi}{6}, ... \end{cases}$. Explain why $f(x)$ is continuous over A. Moreover, define a continuous function $g(x)$ whose domain is B such that $g(x) = f(x)$ for every $x$ in A. Is it possible to extend $f(x)$ on an even larger domain and still be continuous?
