If $f(x) = \begin{cases} \sin x, & \text{if } x \leq 0 \\ x^2+a^2, & \text{if } 0 < x < 1 \\ bx+2, & \text{if } 1 \leq x \leq 2 \\ 0, & \text{if } x > 2 \end{cases}$ is continuous on $\mathbb{R}$, then $a+b+ab = $

If $f(x) = \begin{cases} \frac{\sin((p+1)x) + \sin x}{x} & , x < 0 \\ q & , x = 0 \\ \frac{\sqrt{x+x^2} - \sqrt{x}}{x^{3/2}} & , x > 0 \end{cases}$ is continuous at $x = 0$,then the ordered pair $(p, q)$ is equal to

Let $f: R \rightarrow R$ be defined as $f(x) = \begin{cases} \frac{x^{3}}{(1-\cos 2x)^{2}} \log_{e}\left(\frac{1+2xe^{-2x}}{(1-xe^{-x})^{2}}\right), & x \neq 0 \\ \alpha, & x=0 \end{cases}$. If $f$ is continuous at $x=0$,then $\alpha$ is equal to:

Is the function defined by $f(x) = |x|$ a continuous function?

The values of $A$ and $B$ such that the function $f(x) = \begin{cases} -2\sin x, & x \le -\frac{\pi}{2} \\ A\sin x + B, & -\frac{\pi}{2} < x < \frac{\pi}{2} \\ \cos x, & x \ge \frac{\pi}{2} \end{cases}$ is continuous everywhere are

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 ....... .