The number of points of discontinuity of the function $f(x) = [\frac{x^2}{2}] - [\sqrt{x}]$ for $x \in [0, 4]$,where $[\cdot]$ denotes the greatest integer function,is . . . . . . .

For the function $f(x) = \begin{cases} \frac{\sin^2(ax)}{x^2}, & x \neq 0 \\ 1, & x = 0 \end{cases}$,which one is a true statement?

If $f(x) = \begin{cases} 1 + x^2, & \text{when } 0 \le x \le 1 \\ 1 - x, & \text{when } x > 1 \end{cases}$,then

If $f: R \rightarrow R$ is defined by $f(x) = \begin{cases} \frac{\cos 3x - \cos x}{x^2}, & \text{for } x \neq 0 \\ \lambda, & \text{for } x = 0 \end{cases}$ and if $f$ is continuous at $x = 0$,then $\lambda$ is equal to

The values of $a$ and $b$ such that the function $f(x) = \begin{cases} -2 \sin x, & -\pi \leq x \leq -\frac{\pi}{2} \\ a \sin x + b, & -\frac{\pi}{2} < x < \frac{\pi}{2} \\ \cos x, & \frac{\pi}{2} \leq x \leq \pi \end{cases}$ is continuous in $[-\pi, \pi]$,are

If $f(x) = \log(\sec^2 x)^{\cot^2 x}$ for $x \neq 0$ and $f(x) = K+1$ for $x=0$ is continuous at $x=0$,then the value of $K$ is