Let $[x]$ denote the greatest integer less than or equal to $x$ and $f(x) = [\tan^2 x]$. Which of the following is true?

Let $f: R \rightarrow R$ be defined by $f(x)=\begin{cases} \alpha+\frac{\sin [x]}{x}, & \text{if } x>0 \\ 2, & \text{if } x=0 \\ \beta+\left[\frac{\sin x-x}{x^3}\right], & \text{if } x < 0 \end{cases}$ where $[x]$ denotes the greatest integer function. If $f$ is continuous at $x=0$,then $\beta-\alpha$ is equal to

Discuss the continuity of the function $f$,where $f$ is defined by $f(x) = \begin{cases} -2, & \text{if } x \le -1 \\ 2x, & \text{if } -1 < x \le 1 \\ 2, & \text{if } x > 1 \end{cases}$. Is it continuous at $x=3$?

The value$(s)$ of $x$ for which the function $f(x) = \begin{cases} 1-x, & x < 1 \\ (1-x)(2-x), & 1 \leq x \leq 2 \\ 3-x, & x > 2 \end{cases}$ fails to be continuous is(are):

Let $f(x) = \begin{cases} x, & x \in \mathbb{Q} \\ 1 - x, & x \notin \mathbb{Q} \end{cases}$. Then at $x = \frac{1}{2}$,$f(x)$ is:

If $f:(-7,7) \rightarrow R$ is defined by $f(x)=[x]$ for all $x \in (-7,7)$,then the number of discontinuities of $f$ is