If a function $f$ is defined by $f(x) = \begin{cases} \frac{1-\sqrt{2} \sin x}{\pi-4 x}, & x \neq \frac{\pi}{4} \\ k, & x = \frac{\pi}{4} \end{cases}$ and is continuous at $x = \frac{\pi}{4}$,then $k = $

The set of values of $x$ for which the function $f(x) = \log \left(\frac{x-1}{x+2}\right)$ is continuous, is

Examine whether $f(x) = \sin |x|$ is a continuous function.

If the function $f(x) = \begin{cases} x + a \sqrt{2} \sin x & \text{if } 0 \leq x \leq \frac{\pi}{4} \\ 2x \cot x + b & \text{if } \frac{\pi}{4} < x \leq \frac{\pi}{2} \\ a \cos 2x - b \sin x & \text{if } \frac{\pi}{2} < x \leq \pi \end{cases}$ is continuous in $[0, \pi]$,then $a - b = $

If $f(x) = |x|/x$ for $x \neq 0$ and $1$ for $x = 0$,then the function 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: