If $f(x) = \begin{cases} mx+1, & x \leq \frac{\pi}{2} \\ \sin x+n, & x > \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$,where $m, n \in \mathbb{Z}$,then:

If $f(x) = \begin{cases} \sin^{-1}|x|, & x \ne 0 \\ 0, & x = 0 \end{cases}$ then

If $f(x) = \frac{3^{x+3} - 3^{-x} - 2}{\tan x \cdot \log(1+x)}$ for $x \neq 0$, is continuous at $x = 0$, then the value of $f(0)$ is equal to ...

If $f(x) = \begin{cases} \frac{(8-2x)^{\frac{1}{3}} - 2}{3 - (243+5x)^{\frac{1}{5}}}, & x \neq 0 \\ k, & x = 0 \end{cases}$ is continuous at $x = 0$, then $k =$

If $f(x) = \begin{cases} \frac{\sin [x]}{[x] + 1}, & \text{for } x > 0 \\ \frac{\cos \frac{\pi }{2}[x]}{[x]}, & \text{for } x < 0 \\ k, & \text{at } x = 0 \end{cases}$; where $[x]$ denotes the greatest integer less than or equal to $x$,then in order that $f$ be continuous at $x = 0$,the value of $k$ is

Consider the function $f(x)=\begin{cases} \frac{a(7x-12-x^2)}{b|x^2-7x+12|} & , x<3 \\ 2^{\frac{\sin(x-3)}{x-[x]}} & , x>3 \\ b & , x=3 \end{cases}$ where $[x]$ denotes the greatest integer less than or equal to $x$. If $S$ denotes the set of all ordered pairs $(a, b)$ such that $f(x)$ is continuous at $x=3$,then the number of elements in $S$ is: