Let $f(x) = [x^2] \sin(\pi x)$, for $x > 0$. Then:

If $\lim _{x \rightarrow a^{+}} f(x)=p, \lim _{x \rightarrow a^{-}} f(x)=m$ and $f(a)=k$,then which one of the following is true?

Consider $f(x) = [x] + \sqrt{\{x\}}$,where $[.]$ denotes the greatest integer function and $\{.\}$ denotes the fractional part function. Identify the correct statement.

Let $f(x) = [2x^3 - 5]$,where $[\cdot]$ denotes the Greatest Integer Function. Find the number of points in the interval $(1, 2)$ where the function $f(x)$ is discontinuous.

Let $a, b \in R, (a \ne 0)$. If the function $f$ defined as
$f(x) = \begin{cases} \frac{2x^2}{a}, & 0 \le x < 1 \\ a, & 1 \le x < \sqrt{2} \\ \frac{2b^2 - 4b}{x^3}, & \sqrt{2} \le x < \infty \end{cases}$
is continuous in the interval $[0, \infty)$,then an ordered pair $(a, b)$ is

If $f(x) = \begin{cases} \frac{\sqrt{1 + kx} - \sqrt{1 - kx}}{x} & \text{for } -1 \le x < 0 \\ 2x^2 + 3x - 2 & \text{for } 0 \le x \le 1 \end{cases}$ is continuous at $x = 0$,then $k = $