If the function $f(x) = \begin{cases} \frac{\tan 4x \times \cos 3x}{x} & , x \neq 0 \\ k & , x = 0 \end{cases}$ is continuous at $x = 0$,then the value of $k$ is equal to . . . . . .

Let $a, b \in R, b \neq 0$. Define a function $f(x) = \begin{cases} a \sin \frac{\pi}{2}(x-1), & \text{for } x \leq 0 \\ \frac{\tan 2x - \sin 2x}{bx^3}, & \text{for } x > 0 \end{cases}$. If $f$ is continuous at $x = 0$,then $10 - ab$ is equal to ...... .

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

Let $[t]$ denote the greatest integer $\leq t$. The number of points where the function $f(x)=[x]|x^{2}-1|+\sin \left(\frac{\pi}{[x]+3}\right)-[x+1]$ for $x \in(-2,2)$ is not continuous is:

If $f: R \rightarrow R$ is defined by $f(x) = \begin{cases} x-1, & \text{for } x \leq 1 \\ 2-x^2, & \text{for } 1 < x \leq 3 \\ x-10, & \text{for } 3 < x < 5 \\ 2x, & \text{for } x \geq 5 \end{cases}$,then the set of points of discontinuity of $f$ is

If $f(x) = \begin{cases} \frac{\sqrt{1+ax}-\sqrt{1-ax}}{x}, & -1 \leq x < 0 \\ \frac{x^2+2}{x-2}, & 0 \leq x \leq 1 \end{cases}$ is continuous on $[-1,1]$, then $a=$