If $f(x) = \begin{cases} \frac{1 - \cos x}{x}, & x \ne 0 \\ k, & x = 0 \end{cases}$ is continuous at $x = 0$,then $k = $

Let $f, g: R \rightarrow R$ be functions defined by $f(x) = \begin{cases} x \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases}$ and $g(x) = x f(x)$. Consider the following statements: $(i)$ $f(x)$ is continuous at $x = 0$ but not differentiable at $x = 0$. $(ii)$ $g(x)$ is differentiable at $x = 0$, but $g'(x)$ is not continuous at $x = 0$. Then, which one of the following is true?

If the function $f(x) = \frac{2x - \sin^{-1}x}{2x + \tan^{-1}x}, (x \neq 0)$ is continuous at each point of its domain,then the value of $f(0)$ is

If the function $f(x) = \begin{cases} -2 \sin x, & x \leq \frac{-\pi}{2} \\ A \sin x+B, & \frac{-\pi}{2} < x < \frac{\pi}{2} \\ \cos x, & x \geq \frac{\pi}{2} \end{cases}$ is continuous everywhere,then the values of $A$ and $B$ are respectively

Let $x=2$ be a root of the equation $x^2+px+q=0$ and $f(x)=\begin{cases} \frac{1-\cos(x^2-4px+q^2+8q+16)}{(x-2p)^4}, & x \neq 2p \\ 0, & x=2p \end{cases}$. Then $\lim _{x \rightarrow 2p^{+}}[f(x)]$,where $[.]$ denotes the greatest integer function,is $........$

Let $[\bullet]$ denote the greatest integer function, and let $f(x) = \min \{\sqrt{2}x, x^2\}$. Let $S = \{x \in (-2, 2) : \text{the function } g(x) = |x|[x^2] \text{ is discontinuous at } x\}$. Then $\sum_{x \in S} f(x)$ equals: