If the real valued function $f(x) = \begin{cases} \frac{\cos 3x - \cos x}{x \sin x} & \text{if } x < 0 \\ p & \text{if } x = 0 \\ \frac{\log(1 + q \sin x)}{x} & \text{if } x > 0 \end{cases}$ is continuous at $x = 0$, then $p + q =$

If a function defined by $f(x) = \frac{(3^x - 1)^2}{\sin x \log(1 + x)}$,$x \neq 0$,is continuous at $x = 0$,then $f(0) =$

If $f(x) = \begin{cases} \frac{1-\cos 4x}{x^2} & \text{if } x < 0 \\ a & \text{if } x = 0 \\ \frac{\sqrt{16+\sqrt{x}}-4}{\sqrt{x}} & \text{if } x > 0 \end{cases}$ is continuous at $x = 0$,then find the value of $a$.

Let $f : [a, b] \rightarrow [1, \infty)$ be a continuous function and let $g : \mathbb{R} \rightarrow \mathbb{R}$ be defined as $g(x) = \begin{cases} 0 & \text{if } x < a \\ \int_a^x f(t) dt & \text{if } a \leq x \leq b \\ \int_a^b f(t) dt & \text{if } x > b \end{cases}$. Then:

The number of discontinuities in $\mathbb{R}$ for the function $f(x)=\frac{x-1}{x^3+6x^2+11x+6}$ is

Consider the piecewise defined function $f(x) = \begin{cases} \sqrt{-x} & \text{if } x < 0 \\ 0 & \text{if } 0 \leqslant x \leqslant 4 \\ x - 4 & \text{if } x > 4 \end{cases}$. Choose the answer which best describes the continuity of this function.