If $f(x) = \begin{cases} 1+6x-3x^2, & x \leq 1 \\ x+\log_2(b^2+7), & x > 1 \end{cases}$ is continuous at all real $x$,then $b=$

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 = $

Match the items given in List $A$ with those of the items of List $B$:

$A$. $|x| + |x - 2|$	$I$. Right hand limit does not exist at $x = 2$.
$B$. $\text{cosech } x$	$II$. Continuous only for non-zero real values of $x$.
$C$. $x - [x]$	$III$. Limit is zero for all real $x$.
$D$. $\sqrt{2 - x}$	$IV$. Continuous for all real value of $x$.
	$V$. Discontinuous at all integral values of $x$.

The correct match is:

Show that the function defined by $f(x) = \sin(x^{2})$ is a continuous function.

The function $f(x) = |x - 24|$ is

Let $f: R \rightarrow R$ be a function defined as $f(x)=\begin{cases} \frac{\sin (a+1) x+\sin 2 x}{2 x} & , \text{if } x<0 \\ b & , \text{if } x=0 \\ \frac{\sqrt{x+b x^{3}}-\sqrt{x}}{b x^{5 / 2}} & , \text{if } x>0 \end{cases}$. If $f$ is continuous at $x=0$,then the value of $a+b$ is equal to ....... .