If the function $f(x) = \left[ \frac{(x - 2)^3}{a} \right] \sin(x - 2) + a \cos(x - 2)$ is continuous in $[4, 6]$,then the value of $a$ is (where $[.]$ denotes the greatest integer function).

If $f(x) = \frac{1 - \sin x}{\log(1 + \pi^2 - 4\pi x + 4x^2)}$ is continuous at $x = \frac{\pi}{2}$,then $f\left(\frac{\pi}{2}\right) = $

If $[.]$ denotes the greatest integer function,then $f(x) = [x]^2 - [x^2]$ is discontinuous at

If $f(x) = \begin{cases} \frac{x}{e^{1/x} + 1}, & x \ne 0 \\ 0, & x = 0 \end{cases}$,then

Discuss the continuity of the function $f$,where $f$ is defined by $f(x) = \begin{cases} -2, & \text{if } x \le -1 \\ 2x, & \text{if } -1 < x \le 1 \\ 2, & \text{if } x > 1 \end{cases}$. Is it continuous at $x=3$?

If $f(x) = x^3 + 7x - 1$,then $f(x)$ has a zero between $x = 0$ and $x = 1$. The theorem which best describes this is: