If $f(x) = \begin{cases} \frac{1-\cos 4x}{x^2} & \text{if } x < 0 \\ a & \text{if } x = 0 \\ \frac{\sqrt{16+\sqrt{x}}-4}{\sqrt{x}} & \text{if } x > 0 \end{cases}$ is continuous at $x = 0$,then find the value of $a$.

Let $f:(0, \pi) \rightarrow \mathbb{R}$ be a function given by
$f(x)=\begin{cases} (\frac{8}{7})^{\frac{\tan 8x}{\tan 7x}}, & 0 < x < \frac{\pi}{2} \\ a-8, & x=\frac{\pi}{2} \\ (1+|\cot x|)^{\frac{b}{a}|\tan x|}, & \frac{\pi}{2} < x < \pi \end{cases}$
Where $a, b \in \mathbb{Z}$. If $f$ is continuous at $x=\frac{\pi}{2}$,then $a^2+b^2$ is equal to ..........

Let $f :[0,1] \rightarrow \mathbb{R}$ and $g :[0,1] \rightarrow \mathbb{R}$ be defined as follows:
$f(x) = \begin{cases} 1 & \text{if } x \text{ is rational} \\ 0 & \text{if } x \text{ is irrational} \end{cases}$
$g(x) = \begin{cases} 0 & \text{if } x \text{ is rational} \\ 1 & \text{if } x \text{ is irrational} \end{cases}$
Then:

Let $f(x) = \begin{cases} x^2 + k, & \text{when } x \ge 0 \\ -x^2 - k, & \text{when } x < 0 \end{cases}$. If the function $f(x)$ is continuous at $x = 0$,then $k =$

Consider $f(x) = [x]|x^3 - 2x^2 - x + 2|$ in $[-\frac{3}{2}, \frac{9}{2}]$. The number of points where $f(x)$ is discontinuous is (where $[.]$ denotes the greatest integer function).

Given $f(x) = \begin{cases} \frac{\ln(1+\text{sgn}[x]+{x}^2)}{1-\cos{x}} & \text{if } x \neq 0 \\ k & \text{if } x = 0 \end{cases}$ (where $[\cdot]$,${\cdot}$ and $\text{sgn } x$ denote the greatest integer function,fractional part function,and signum function respectively),which of the following is true?