If $f(x) = \operatorname{sgn} \left( 3\cos x - \frac{a}{3} \right)$ is continuous for all $x$,then the least positive integral value of $a$ is - (where $\operatorname{sgn}(x)$ denotes the signum function of $x$)

The function $f(x) = \sin |x|$ is

Let $f(x) = \begin{cases} \alpha + \frac{\sin [x]}{x}, & \text{if } x > 0 \\ 2, & \text{if } x = 0 \\ \beta + \left[ \frac{\sin x - x}{x^3} \right], & \text{if } x < 0 \end{cases}$ where $[x]$ denotes the greatest integer function. If $f$ is continuous at $x = 0$, then $\beta - \alpha$ is equal to

Prove that the function $f(x)=5x-3$ is continuous at $x=0$,$x=-3$,and $x=5$.

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 the function $f(x) = \frac{1-\sin 2x + \cos 2x}{1+\sin 2x + \cos 2x}$ for $x \neq \frac{\pi}{2}$ and $f(x) = k$ for $x = \frac{\pi}{2}$ is continuous at $x = \frac{\pi}{2}$,then $k = $