If the function $f(x) = \frac{4\sqrt{2}(\sin 3x + \sin x)}{2 \sin 2x \sin \frac{3x}{2} + \cos \frac{5x}{2} - \cos \frac{3x}{2}}$ for $x \neq \frac{\pi}{2}$ is continuous at $x = \frac{\pi}{2}$, then the value of $f(\frac{\pi}{2})$ is equal to

Find all points of discontinuity of the function $f$ defined by $f(x) = |x| - |x + 1|$.

Let $f$ be a differentiable function on the open interval $(a, b)$. Which of the following statements must be true?
$I$. $f$ is continuous on the closed interval $[a, b]$
$II$. $f$ is bounded on the open interval $(a, b)$
$III$. If $a < a_1 < b_1 < b$,and $f(a_1) < 0 < f(b_1)$,then there is a number $c$ such that $a_1 < c < b_1$ and $f(c) = 0$

If the function $f(x) = \frac{e^{x}(e^{\tan x-x}-1)+\log_{e}(\sec x+\tan x)-x}{\tan x-x}$ is continuous at $x=0$, then the value of $f(0)$ is equal to

If the function $f(x) = \begin{cases} 3ax + b, & \text{for } x < 1 \\ 11, & \text{for } x = 1 \\ 5ax - 2b, & \text{for } x > 1 \end{cases}$ is continuous at $x = 1$,then the values of $a$ and $b$ are:

The function $f(x) = \frac{2x^2 + 7}{x^3 + 3x^2 - x - 3}$ is discontinuous for